💡First, I need to determine the total amount Mitzi spent in yen. She spent 3,000 yen on a Tokyo Disneyland ticket, 2,500 yen on food, and 1,500 yen on a T-shirt. Additionally, she bought a souvenir originally priced at 2,200 yen with a 20% discount.To find the discounted price of the souvenir, I'll calculate 20% of 2,200 yen and subtract it from the original price:20% of 2,200 yen = 0.20 * 2,200 yen = 440 yenDiscounted price of the souvenir = 2,200 yen - 440 yen = 1,760 yenNow, I'll add up all the expenses:Total expenses = 3,000 yen + 2,500 yen + 1,500 yen + 1,760 yen = 8,760 yenMitzi started with 10,000 yen, so the amount she has left is:10,000 yen - 8,760 yen = 1,240 yenFinally, I'll convert the remaining yen to dollars using the exchange rate of 110 yen per dollar:1,240 yen / 110 yen/dollar ≈ 11.2727 dollarsRounded to the nearest cent, Mitzi could spend approximately 11.27 at the store.

💡Okay, so I have this problem where there are five points, A, B, C, D, and E, arranged equally spaced on a circle. I'm given two vectors: vector DA is equal to vector a, and vector DB is equal to vector b. I need to express the vector AC in terms of vectors a and b.First, I need to visualize the setup. Since the points are equally spaced on a circle, the angle between each adjacent point should be the same. There are five points, so the circle is divided into five equal arcs. That means the central angle between each point is 360 degrees divided by 5, which is 72 degrees. So, each point is 72 degrees apart from the next.Let me try to sketch this mentally. Let's say point D is at the bottom of the circle. Then moving counterclockwise, the next point is E, then A, then B, then C, and back to D. So, the order is D, E, A, B, C, D.Given that, vector DA is from D to A, and vector DB is from D to B. So, DA is the vector pointing from D to A, and DB is the vector pointing from D to B.I need to find vector AC, which is from A to C. So, starting at A and ending at C.Since all points are on a circle, I can think of each vector as a position vector from the center of the circle. But since vectors DA and DB are given, maybe I can express AC in terms of these vectors.Let me denote the center of the circle as O. Then, vectors DA and DB can be expressed in terms of position vectors from O. So, vector DA would be OA - OD, and vector DB would be OB - OD.But I'm not sure if that's the right approach. Maybe I should consider the vectors DA and DB as vectors from point D to points A and B, respectively. So, DA is the vector from D to A, and DB is the vector from D to B.Since the points are equally spaced, the angle between DA and DB is 72 degrees because from D to A is one step, and from D to B is two steps, so the angle between them is 72 degrees times 2, which is 144 degrees? Wait, no. From D to A is one step, which is 72 degrees, and from D to B is two steps, which would be 144 degrees. So, the angle between vectors DA and DB is 144 degrees.Hmm, maybe I can use vector addition or some trigonometric relations to express AC in terms of DA and DB.Alternatively, since all points are on a circle, maybe I can use complex numbers to represent the points, treating the circle as the unit circle in the complex plane. That might make it easier to handle rotations and vector additions.Let me try that approach. Let's assign complex numbers to each point. Let’s assume the circle has a radius of 1 for simplicity, and let’s place point D at (1, 0) on the complex plane. Then, moving counterclockwise, each point is spaced by 72 degrees.So, point D is at angle 0 degrees, point E is at 72 degrees, point A is at 144 degrees, point B is at 216 degrees, and point C is at 288 degrees.Wait, actually, if D is at 0 degrees, then moving counterclockwise, the next point E would be at 72 degrees, then A at 144 degrees, B at 216 degrees, and C at 288 degrees, and back to D at 360 degrees, which is the same as 0 degrees.So, in complex numbers, point D is 1 (cos0 + isin0), point E is e^(i72°), point A is e^(i144°), point B is e^(i216°), and point C is e^(i288°).Given that, vector DA is from D to A, which would be A - D in complex numbers. Similarly, vector DB is B - D.So, vector DA = A - D = e^(i144°) - 1.Vector DB = B - D = e^(i216°) - 1.I need to express vector AC, which is C - A = e^(i288°) - e^(i144°).So, I need to express e^(i288°) - e^(i144°) in terms of e^(i144°) - 1 and e^(i216°) - 1.Hmm, this seems a bit complicated. Maybe I can express e^(i288°) in terms of e^(i144°) and e^(i216°).Wait, 288 degrees is 144 degrees plus 144 degrees, but that's 288 degrees, which is correct. Alternatively, 288 degrees is 360 - 72 degrees, which is equivalent to -72 degrees.But I'm not sure if that helps. Maybe I can use the fact that e^(iθ) = cosθ + i sinθ.Let me compute the vectors:First, let's compute DA and DB.DA = e^(i144°) - 1.DB = e^(i216°) - 1.AC = e^(i288°) - e^(i144°).I need to express AC in terms of DA and DB.Let me see if I can find a linear combination of DA and DB that equals AC.So, suppose AC = k*DA + m*DB, where k and m are scalars.Then, e^(i288°) - e^(i144°) = k*(e^(i144°) - 1) + m*(e^(i216°) - 1).I need to solve for k and m.This seems like a system of equations, but since we're dealing with complex numbers, it's a bit more involved.Alternatively, maybe I can express e^(i288°) in terms of e^(i144°) and e^(i216°).Note that 288° = 144° + 144°, so e^(i288°) = (e^(i144°))^2.Similarly, e^(i216°) = e^(i144° + i72°) = e^(i144°)*e^(i72°).But I'm not sure if that helps directly.Alternatively, maybe I can use the fact that the points are equally spaced, so the vectors DA, DB, etc., have certain symmetries.Wait, another approach: since all points are on a circle, the vectors from the center can be used to express the position vectors, and then the vectors DA and DB can be expressed in terms of these.Let me denote the position vectors of points A, B, C, D, E as vectors OA, OB, OC, OD, OE, respectively.Given that, vector DA = OA - OD, and vector DB = OB - OD.I need to find vector AC = OC - OA.So, AC = OC - OA.I need to express OC in terms of OA and OB, or in terms of DA and DB.But I have DA = OA - OD, so OA = DA + OD.Similarly, DB = OB - OD, so OB = DB + OD.But I don't have OD directly. Maybe I can express OD in terms of OA or OB.Alternatively, since all points are equally spaced, the vectors OA, OB, OC, OD, OE are all rotations of each other by 72 degrees.So, if I let vector OD be some vector, say, d, then OA is d rotated by 144 degrees, OB is d rotated by 216 degrees, and so on.But I'm not sure if that helps directly.Wait, maybe I can use the fact that the points are equally spaced to express OC in terms of OA and OB.Since from A to B is 72 degrees, and from B to C is another 72 degrees, so from A to C is 144 degrees.But I'm not sure.Alternatively, maybe I can use the law of cosines on the vectors.Given that, the angle between DA and DB is 144 degrees, as I thought earlier.So, if I consider triangle DAB, I can use the law of cosines to relate DA, DB, and AB.But I'm not sure if that helps with AC.Wait, maybe I can express AC as a combination of DA and DB.Let me think geometrically. If I have vectors DA and DB, can I somehow combine them to get AC?Alternatively, maybe I can express AC in terms of OA and OC, and then express OA and OC in terms of DA and DB.Wait, OA = DA + OD, and OC = OD + DC.But I don't know vector DC.Alternatively, maybe I can express OC in terms of OA and some rotation.Since from A to C is two steps, which is 144 degrees, so OC is OA rotated by 144 degrees.But I don't know if that helps in terms of DA and DB.Wait, maybe I can express OA and OB in terms of DA and DB, and then express OC in terms of OA and OB.Let me try that.Given DA = OA - OD, so OA = DA + OD.Similarly, DB = OB - OD, so OB = DB + OD.But I don't know OD. Maybe I can express OD in terms of OA or OB.Wait, if I consider that OA, OB, OC, OD, OE are all vectors from the center, and they are equally spaced, then the angle between OA and OB is 72 degrees, and so on.But I'm not sure.Alternatively, maybe I can express OD in terms of OA.Since OA is 144 degrees from OD, so OD = OA rotated by -144 degrees.But I'm not sure.Wait, maybe I can use complex numbers again, but this time express everything in terms of DA and DB.Let me denote DA as vector a and DB as vector b.So, a = OA - OD, and b = OB - OD.I need to find AC = OC - OA.So, AC = OC - OA.But I need to express OC in terms of OA and OB, or in terms of a and b.Wait, maybe I can express OC as OA + 2*(AB), since from A to B is AB, and from B to C is another AB, so AC = AB + AB = 2AB.But AB is OB - OA.So, AC = 2*(OB - OA).But OB = DB + OD, and OA = a + OD.So, AC = 2*( (DB + OD) - (a + OD) ) = 2*(DB - a).But DB is vector b, so AC = 2*(b - a).Wait, is that correct? Let me check.If AC = 2*(OB - OA), and OB - OA = AB, then AC = 2AB.But AB is from A to B, which is vector DB - DA, because DB is from D to B, and DA is from D to A, so AB = DB - DA.Wait, no, AB is OB - OA, which is (DB + OD) - (DA + OD) = DB - DA.So, AB = DB - DA = b - a.Therefore, AC = 2*(b - a).Wait, but that seems too straightforward. Let me verify.If AC is from A to C, which is two steps from A, so it's equivalent to two AB vectors.So, AC = AB + AB = 2AB.Since AB = OB - OA = (DB + OD) - (DA + OD) = DB - DA = b - a.Therefore, AC = 2*(b - a).But wait, in the complex plane, if I compute AC as e^(i288°) - e^(i144°), and compute 2*(b - a), where a = e^(i144°) - 1 and b = e^(i216°) - 1, does it match?Let me compute 2*(b - a):2*(b - a) = 2*( (e^(i216°) - 1) - (e^(i144°) - 1) ) = 2*(e^(i216°) - e^(i144°)).But AC is e^(i288°) - e^(i144°).So, 2*(b - a) = 2*(e^(i216°) - e^(i144°)).Is this equal to e^(i288°) - e^(i144°)?Let me compute e^(i288°):e^(i288°) = cos288° + i sin288° = cos(-72°) + i sin(-72°) = cos72° - i sin72°.Similarly, e^(i216°) = cos216° + i sin216° = cos(180°+36°) + i sin(180°+36°) = -cos36° - i sin36°.And e^(i144°) = cos144° + i sin144° = cos(180°-36°) + i sin(180°-36°) = -cos36° + i sin36°.So, 2*(e^(i216°) - e^(i144°)) = 2*( (-cos36° - i sin36°) - (-cos36° + i sin36°) ) = 2*( -cos36° - i sin36° + cos36° - i sin36° ) = 2*( -2i sin36° ) = -4i sin36°.On the other hand, AC = e^(i288°) - e^(i144°) = (cos72° - i sin72°) - (-cos36° + i sin36°) = cos72° + cos36° - i sin72° - i sin36°.Compute cos72° + cos36°:cos72° ≈ 0.3090, cos36° ≈ 0.8090, so sum ≈ 1.1180.Similarly, sin72° ≈ 0.9511, sin36° ≈ 0.5878, so sum ≈ 1.5389.So, AC ≈ 1.1180 - i*1.5389.But 2*(b - a) = -4i sin36° ≈ -4i*0.5878 ≈ -2.3512i.These are not equal. So, my previous conclusion that AC = 2*(b - a) is incorrect.Hmm, so my approach was wrong. Maybe I need to consider a different method.Let me think again. Since all points are on a circle, the vectors DA and DB can be used to express other vectors through rotations.Since each step is 72 degrees, rotating a vector by 72 degrees would give the next vector.So, if I rotate vector DA by 72 degrees, I should get vector DB.Wait, is that true?Vector DA is from D to A, which is 144 degrees from D. Rotating DA by 72 degrees counterclockwise would point towards B, so yes, vector DB is vector DA rotated by 72 degrees.So, in complex numbers, if DA is a vector, then DB = DA * e^(i72°).Similarly, AC can be expressed as a rotation of some vector.But I need to express AC in terms of DA and DB, which are given.Alternatively, maybe I can express AC as a combination of DA and DB using some coefficients.Let me denote AC = k*DA + m*DB, where k and m are scalars to be determined.So, AC = k*a + m*b.I need to find k and m such that this holds.But how?Well, in complex numbers, I can write the equation:e^(i288°) - e^(i144°) = k*(e^(i144°) - 1) + m*(e^(i216°) - 1).This is an equation in complex numbers, so both the real and imaginary parts must match.Let me compute each term.First, compute e^(i288°) - e^(i144°):As before, e^(i288°) = cos72° - i sin72° ≈ 0.3090 - i*0.9511.e^(i144°) = cos144° + i sin144° ≈ -0.8090 + i*0.5878.So, e^(i288°) - e^(i144°) ≈ (0.3090 - (-0.8090)) + i*(-0.9511 - 0.5878) ≈ 1.1180 - i*1.5389.Now, compute k*(e^(i144°) - 1) + m*(e^(i216°) - 1):First, e^(i144°) - 1 ≈ (-0.8090 - 1) + i*0.5878 ≈ -1.8090 + i*0.5878.Similarly, e^(i216°) - 1 ≈ (-0.8090 - 1) + i*(-0.5878) ≈ -1.8090 - i*0.5878.So, the equation becomes:1.1180 - i*1.5389 = k*(-1.8090 + i*0.5878) + m*(-1.8090 - i*0.5878).Let me write this as:(1.1180) + (-1.5389)i = (-1.8090k - 1.8090m) + (0.5878k - 0.5878m)i.Now, equate the real and imaginary parts:Real: 1.1180 = -1.8090(k + m).Imaginary: -1.5389 = 0.5878(k - m).So, we have two equations:1) -1.8090(k + m) = 1.1180.2) 0.5878(k - m) = -1.5389.Let me solve equation 2 first:0.5878(k - m) = -1.5389.Divide both sides by 0.5878:k - m = -1.5389 / 0.5878 ≈ -2.6180.Similarly, equation 1:-1.8090(k + m) = 1.1180.Divide both sides by -1.8090:k + m = 1.1180 / (-1.8090) ≈ -0.6180.So, now we have:k - m ≈ -2.6180,k + m ≈ -0.6180.Let me add these two equations:(k - m) + (k + m) = -2.6180 + (-0.6180),2k = -3.2360,k ≈ -1.6180.Then, from equation 2:k - m ≈ -2.6180,-1.6180 - m ≈ -2.6180,-m ≈ -2.6180 + 1.6180 ≈ -1.0000,m ≈ 1.0000.So, k ≈ -1.6180, m ≈ 1.0000.But these numbers look familiar. 1.6180 is approximately the golden ratio, (1 + sqrt(5))/2 ≈ 1.6180.So, k ≈ - (1 + sqrt(5))/2 ≈ -1.6180,m ≈ 1.Therefore, AC ≈ - (1 + sqrt(5))/2 * DA + 1 * DB.So, AC = DB - (1 + sqrt(5))/2 * DA.Since DA is vector a and DB is vector b, we can write:AC = b - (1 + sqrt(5))/2 * a.Let me check if this makes sense.If I plug in the values:AC ≈ b - 1.6180a.Given that a = DA ≈ e^(i144°) - 1 ≈ -1.8090 + i*0.5878,and b = DB ≈ e^(i216°) - 1 ≈ -1.8090 - i*0.5878,then 1.6180a ≈ 1.6180*(-1.8090 + i*0.5878) ≈ -2.9280 + i*0.9520.So, b - 1.6180a ≈ (-1.8090 - i*0.5878) - (-2.9280 + i*0.9520) ≈ (-1.8090 + 2.9280) + (-0.5878 - 0.9520)i ≈ 1.1190 - i*1.5398.Which is very close to the computed AC ≈ 1.1180 - i*1.5389.So, this seems correct.Therefore, the vector AC can be expressed as:AC = b - (1 + sqrt(5))/2 * a.So, in terms of vectors a and b, AC is equal to vector b minus (1 + sqrt(5))/2 times vector a.I think that's the answer.

💡Alright, so I have this problem about finding the locus of the centers of circles that are externally tangent to one circle and internally tangent to another. The circles given are ( C_1 ) with equation ( x^2 + y^2 = 1 ) and ( C_2 ) with equation ( (x - 2)^2 + y^2 = 25 ). I need to find the equation of the locus of centers ( (a, b) ) of all such circles.First, let me visualize the problem. ( C_1 ) is a circle centered at the origin (0,0) with radius 1, and ( C_2 ) is a circle centered at (2,0) with radius 5. So, ( C_2 ) is much larger and shifted 2 units to the right along the x-axis.Now, I need to find circles that are externally tangent to ( C_1 ) and internally tangent to ( C_2 ). Let's denote the center of such a circle as ( (a, b) ) and its radius as ( r ).Since the circle is externally tangent to ( C_1 ), the distance between their centers should be equal to the sum of their radii. The center of ( C_1 ) is at (0,0), so the distance between (a, b) and (0,0) is ( sqrt{a^2 + b^2} ). This should equal ( r + 1 ) because of external tangency. So, I can write the equation:[ sqrt{a^2 + b^2} = r + 1 ]Similarly, since the circle is internally tangent to ( C_2 ), the distance between their centers should be equal to the difference of their radii. The center of ( C_2 ) is at (2,0), so the distance between (a, b) and (2,0) is ( sqrt{(a - 2)^2 + b^2} ). This should equal ( 5 - r ) because of internal tangency. So, I can write another equation:[ sqrt{(a - 2)^2 + b^2} = 5 - r ]Now, I have two equations:1. ( sqrt{a^2 + b^2} = r + 1 )2. ( sqrt{(a - 2)^2 + b^2} = 5 - r )I need to eliminate ( r ) to find the relationship between ( a ) and ( b ). Let me square both equations to make them easier to handle.Starting with the first equation:[ (sqrt{a^2 + b^2})^2 = (r + 1)^2 ][ a^2 + b^2 = r^2 + 2r + 1 ] ...(1)And the second equation:[ (sqrt{(a - 2)^2 + b^2})^2 = (5 - r)^2 ][ (a - 2)^2 + b^2 = 25 - 10r + r^2 ] ...(2)Now, subtract equation (1) from equation (2) to eliminate ( r^2 ):[ [(a - 2)^2 + b^2] - [a^2 + b^2] = (25 - 10r + r^2) - (r^2 + 2r + 1) ]Simplify both sides:Left side:[ (a^2 - 4a + 4 + b^2) - (a^2 + b^2) ][ = -4a + 4 ]Right side:[ 25 - 10r + r^2 - r^2 - 2r - 1 ][ = 24 - 12r ]So, we have:[ -4a + 4 = 24 - 12r ]Let me solve for ( r ):First, subtract 24 from both sides:[ -4a + 4 - 24 = -12r ][ -4a - 20 = -12r ]Divide both sides by -4:[ a + 5 = 3r ][ r = frac{a + 5}{3} ]Wait, let me check that step again. When I subtract 24, it's:[ -4a + 4 - 24 = -12r ][ -4a - 20 = -12r ]Divide both sides by -4:[ a + 5 = 3r ][ r = frac{a + 5}{3} ]Yes, that seems correct.Now, substitute ( r = frac{a + 5}{3} ) back into equation (1):[ a^2 + b^2 = left( frac{a + 5}{3} + 1 right)^2 ]Simplify the right side:First, ( frac{a + 5}{3} + 1 = frac{a + 5 + 3}{3} = frac{a + 8}{3} )So, equation (1) becomes:[ a^2 + b^2 = left( frac{a + 8}{3} right)^2 ][ a^2 + b^2 = frac{(a + 8)^2}{9} ]Multiply both sides by 9 to eliminate the denominator:[ 9a^2 + 9b^2 = (a + 8)^2 ]Expand the right side:[ 9a^2 + 9b^2 = a^2 + 16a + 64 ]Bring all terms to the left side:[ 9a^2 + 9b^2 - a^2 - 16a - 64 = 0 ][ 8a^2 + 9b^2 - 16a - 64 = 0 ]Hmm, that's not the same as the initial thought process. Wait, let me check my steps again.Wait, in the initial thought process, I had:After subtracting equations, I got:[ 4a - 4 = 2r - 24 ]Which led to:[ r = 2a + 10 ]But in my current calculation, I have:After subtracting, I got:[ -4a + 4 = 24 - 12r ]Which led to:[ -4a - 20 = -12r ][ a + 5 = 3r ][ r = frac{a + 5}{3} ]So, there's a discrepancy here. Let me see where I went wrong.Wait, in the initial thought process, when subtracting the equations, the user had:( a^2 - (a - 2)^2 = (r + 1)^2 - (5 - r)^2 )Let me compute both sides:Left side:( a^2 - (a^2 - 4a + 4) = 4a - 4 )Right side:( (r + 1)^2 - (5 - r)^2 )Expand both:( r^2 + 2r + 1 - (25 - 10r + r^2) )Simplify:( r^2 + 2r + 1 - 25 + 10r - r^2 )( 12r - 24 )So, equation becomes:( 4a - 4 = 12r - 24 )Which simplifies to:( 4a - 4 = 12r - 24 )Subtract 12r from both sides:( 4a - 12r - 4 = -24 )Add 24 to both sides:( 4a - 12r + 20 = 0 )Divide by 4:( a - 3r + 5 = 0 )So, ( a + 5 = 3r )Thus, ( r = frac{a + 5}{3} )Wait, so in the initial thought process, the user had:After subtracting, got ( 4a - 4 = 2r - 24 ), which led to ( r = 2a + 10 ). But that seems incorrect because the right side should be ( 12r - 24 ), not ( 2r - 24 ). So, the initial thought process had a mistake in the subtraction step.So, in my current calculation, I have ( r = frac{a + 5}{3} ), which seems correct.Now, substituting ( r = frac{a + 5}{3} ) into equation (1):[ a^2 + b^2 = left( frac{a + 5}{3} + 1 right)^2 ][ = left( frac{a + 5 + 3}{3} right)^2 ][ = left( frac{a + 8}{3} right)^2 ][ = frac{(a + 8)^2}{9} ]Multiply both sides by 9:[ 9a^2 + 9b^2 = (a + 8)^2 ][ 9a^2 + 9b^2 = a^2 + 16a + 64 ]Bring all terms to the left:[ 9a^2 + 9b^2 - a^2 - 16a - 64 = 0 ][ 8a^2 + 9b^2 - 16a - 64 = 0 ]Hmm, this is a quadratic equation, but it's not in the standard form. Let me see if I can simplify it further or write it in the required form.The problem asks for the equation in the form ( Pa^2 + Qb^2 + Ra + Sb + T = 0 ) with integer coefficients, P positive, and gcd 1.Looking at my current equation:[ 8a^2 + 9b^2 - 16a - 64 = 0 ]This is already in the required form, with P=8, Q=9, R=-16, S=0, T=-64.But let me check if I can simplify it further by dividing by a common factor. The coefficients are 8, 9, -16, 0, -64.The gcd of 8,9,16,64 is 1, since 8 and 9 are coprime. So, the equation is already in the simplest form.Wait, but in the initial thought process, the user had a different equation: ( 3a^2 + b^2 + 44a + 121 = 0 ). So, clearly, there's a mistake somewhere in my calculation.Let me go back and check.Wait, in my substitution step, I had:From equation (1):[ a^2 + b^2 = left( frac{a + 8}{3} right)^2 ][ 9a^2 + 9b^2 = (a + 8)^2 ][ 9a^2 + 9b^2 = a^2 + 16a + 64 ][ 8a^2 + 9b^2 - 16a - 64 = 0 ]This seems correct.But in the initial thought process, the user had:After substituting ( r = 2a + 10 ) into equation (1):[ a^2 + b^2 = (2a + 11)^2 ][ a^2 + b^2 = 4a^2 + 44a + 121 ][ 3a^2 + 44a + b^2 + 121 = 0 ][ 3a^2 + b^2 + 44a + 121 = 0 ]But in my calculation, I have ( r = frac{a + 5}{3} ), which is different from the user's ( r = 2a + 10 ). So, the mistake must be in the initial subtraction step.Wait, let me re-examine the subtraction step.From the two equations:1. ( a^2 + b^2 = r^2 + 2r + 1 )2. ( (a - 2)^2 + b^2 = 25 - 10r + r^2 )Subtracting equation (1) from equation (2):[ (a - 2)^2 + b^2 - a^2 - b^2 = 25 - 10r + r^2 - r^2 - 2r - 1 ]Simplify left side:[ (a^2 - 4a + 4 + b^2) - a^2 - b^2 = -4a + 4 ]Right side:[ 25 - 10r - 2r - 1 = 24 - 12r ]So, equation becomes:[ -4a + 4 = 24 - 12r ][ -4a = 20 - 12r ][ -4a + 12r = 20 ]Divide by 4:[ -a + 3r = 5 ][ 3r = a + 5 ][ r = frac{a + 5}{3} ]Yes, that's correct. So, the initial thought process had an error in the subtraction step, leading to an incorrect value of ( r ).Therefore, my current equation ( 8a^2 + 9b^2 - 16a - 64 = 0 ) is correct.But let me check if this can be simplified or if I made a mistake in substitution.Wait, let me try another approach. Maybe instead of subtracting the equations, I can express ( r ) from one equation and substitute into the other.From equation (1):[ sqrt{a^2 + b^2} = r + 1 ]So, ( r = sqrt{a^2 + b^2} - 1 )From equation (2):[ sqrt{(a - 2)^2 + b^2} = 5 - r ]So, ( r = 5 - sqrt{(a - 2)^2 + b^2} )Set them equal:[ sqrt{a^2 + b^2} - 1 = 5 - sqrt{(a - 2)^2 + b^2} ][ sqrt{a^2 + b^2} + sqrt{(a - 2)^2 + b^2} = 6 ]Hmm, this is the equation of an ellipse, where the sum of distances from two foci is constant. The foci are at (0,0) and (2,0), and the sum is 6.But wait, the distance between the two foci is 2 units (from (0,0) to (2,0)). For an ellipse, the sum of distances must be greater than the distance between the foci. Here, 6 > 2, so it is an ellipse.But in our case, the circles are externally tangent to ( C_1 ) and internally tangent to ( C_2 ). So, the locus should be an ellipse.But according to my previous calculation, I have a quadratic equation, which is an ellipse. Let me see if my equation ( 8a^2 + 9b^2 - 16a - 64 = 0 ) can be written in standard form.Let me complete the squares.First, group terms:[ 8a^2 - 16a + 9b^2 = 64 ]Factor out coefficients of squared terms:[ 8(a^2 - 2a) + 9b^2 = 64 ]Complete the square for ( a ):( a^2 - 2a = (a - 1)^2 - 1 )So,[ 8[(a - 1)^2 - 1] + 9b^2 = 64 ][ 8(a - 1)^2 - 8 + 9b^2 = 64 ][ 8(a - 1)^2 + 9b^2 = 72 ]Divide both sides by 72 to get standard form:[ frac{(a - 1)^2}{9} + frac{b^2}{8} = 1 ]Yes, this is an ellipse centered at (1, 0) with semi-major axis 3 along the x-axis and semi-minor axis ( 2sqrt{2} ) along the y-axis.But the problem asks for the equation in the form ( Pa^2 + Qb^2 + Ra + Sb + T = 0 ). So, expanding the standard form:[ frac{(a - 1)^2}{9} + frac{b^2}{8} = 1 ]Multiply both sides by 72:[ 8(a - 1)^2 + 9b^2 = 72 ]Expand ( (a - 1)^2 ):[ 8(a^2 - 2a + 1) + 9b^2 = 72 ][ 8a^2 - 16a + 8 + 9b^2 = 72 ]Bring constants to the right:[ 8a^2 + 9b^2 - 16a + 8 - 72 = 0 ][ 8a^2 + 9b^2 - 16a - 64 = 0 ]Which matches my earlier equation. So, this is correct.But wait, in the initial thought process, the user had a different equation, which was incorrect because of a mistake in the subtraction step. So, the correct equation is ( 8a^2 + 9b^2 - 16a - 64 = 0 ).But let me check if this can be simplified further or if I made a mistake in the process.Alternatively, let me consider the initial thought process again. The user had:After subtracting, got ( 4a - 4 = 2r - 24 ), leading to ( r = 2a + 10 ). Then substituted into equation (1):[ a^2 + b^2 = (2a + 11)^2 ]Which led to ( 3a^2 + b^2 + 44a + 121 = 0 ).But as we saw, the correct value of ( r ) is ( frac{a + 5}{3} ), not ( 2a + 10 ). So, the initial thought process had an error.Therefore, the correct equation is ( 8a^2 + 9b^2 - 16a - 64 = 0 ).But let me check if this can be simplified or if I made a mistake in the process.Wait, another way to approach this is to consider the definition of the ellipse. The sum of distances from two points is constant. In this case, the sum is 6, as we saw earlier.But in our case, the sum of distances is 6, and the distance between the foci is 2. So, the major axis length is 6, so semi-major axis ( a = 3 ), and the distance between center and each focus is ( c = 1 ) (since the foci are at (0,0) and (2,0), the center is at (1,0), and the distance from center to each focus is 1).Then, the semi-minor axis ( b ) can be found using ( c^2 = a^2 - b^2 ).Wait, no, actually, in an ellipse, ( c^2 = a^2 - b^2 ), where ( c ) is the distance from center to focus, ( a ) is semi-major axis, ( b ) is semi-minor axis.But in our case, the sum of distances is 2a = 6, so a = 3. The distance between foci is 2c = 2, so c = 1.Then, ( b^2 = a^2 - c^2 = 9 - 1 = 8 ). So, the standard form of the ellipse is:[ frac{(x - h)^2}{a^2} + frac{(y - k)^2}{b^2} = 1 ]Where (h,k) is the center. The center is the midpoint between (0,0) and (2,0), which is (1,0). So, the equation is:[ frac{(x - 1)^2}{9} + frac{y^2}{8} = 1 ]Which, when expanded, is:[ 8(x - 1)^2 + 9y^2 = 72 ][ 8x^2 - 16x + 8 + 9y^2 = 72 ][ 8x^2 + 9y^2 - 16x - 64 = 0 ]Which matches my earlier result. So, this confirms that the correct equation is ( 8a^2 + 9b^2 - 16a - 64 = 0 ).But wait, the problem asks for the equation in the form ( Pa^2 + Qb^2 + Ra + Sb + T = 0 ). So, my equation is already in that form, with P=8, Q=9, R=-16, S=0, T=-64.But let me check if I can simplify it further by dividing by a common factor. The coefficients are 8, 9, -16, 0, -64. The gcd of these coefficients is 1, since 8 and 9 are coprime. So, the equation is already in the simplest form.Therefore, the locus of the centers is given by the equation:[ 8a^2 + 9b^2 - 16a - 64 = 0 ]But let me double-check my steps once more to ensure there are no mistakes.1. Defined the problem correctly.2. Set up the two distance equations correctly.3. Squared both equations correctly.4. Subtracted the equations correctly, leading to ( r = frac{a + 5}{3} ).5. Substituted ( r ) back into equation (1) correctly.6. Expanded and simplified correctly to get ( 8a^2 + 9b^2 - 16a - 64 = 0 ).7. Verified by considering the ellipse definition and standard form, which matches the result.Yes, everything seems correct. So, the final answer is ( 8a^2 + 9b^2 - 16a - 64 = 0 ).

💡Okay, so I have this problem about a line given by the equation (2 + m)x + (1 - 2m)y + (4 - 3m) = 0. There are two parts: first, I need to prove that no matter what value m takes, the line always passes through a fixed point M. Second, I have to find the equation of another line l₁ that goes through M and forms a triangle with the negative semi-axes of the coordinate axes, and this triangle should have the smallest possible area. Hmm, okay, let's tackle the first part first.For part (1), I remember that if a line passes through a fixed point regardless of the parameter, then the equation must hold true for all values of m. So, maybe I can rearrange the equation to group the terms involving m together. Let me try that.Starting with the given equation:(2 + m)x + (1 - 2m)y + (4 - 3m) = 0I can expand this to:2x + mx + y - 2my + 4 - 3m = 0Now, let's collect the terms with m:mx - 2my - 3m + 2x + y + 4 = 0Factor out m from those terms:m(x - 2y - 3) + 2x + y + 4 = 0So, the equation becomes:m(x - 2y - 3) + (2x + y + 4) = 0Now, this equation must hold true for all values of m. That means the coefficient of m must be zero, and the constant term must also be zero. Otherwise, for different m, the equation would represent different lines. So, setting the coefficients equal to zero:1. Coefficient of m: x - 2y - 3 = 02. Constant term: 2x + y + 4 = 0So now, I have a system of two equations:x - 2y - 3 = 02x + y + 4 = 0I need to solve this system to find the fixed point M.Let me write them again:1. x - 2y = 32. 2x + y = -4I can solve this using substitution or elimination. Let's try elimination. Maybe multiply the first equation by 2 to make the coefficients of x the same.Multiplying equation 1 by 2:2x - 4y = 6Now, subtract equation 2 from this:(2x - 4y) - (2x + y) = 6 - (-4)2x - 4y - 2x - y = 10-5y = 10So, y = -2Now, plug y = -2 into equation 2:2x + (-2) = -42x - 2 = -42x = -2x = -1So, the fixed point M is (-1, -2). That makes sense. So, regardless of m, the line passes through (-1, -2). I think that settles part (1).Now, moving on to part (2). I need to draw a line l₁ through M that forms a triangle with the negative semi-axes, and this triangle should have the smallest possible area. Hmm, okay.First, let's visualize this. The negative semi-axes are the negative x-axis and negative y-axis. So, the triangle will be in the third quadrant, right? Because the negative semi-axes meet at the origin, and the line l₁ passes through M(-1, -2). So, the line will intersect both the negative x-axis and negative y-axis, forming a triangle with the origin.To find the equation of line l₁, I need to find its slope and intercept. Let me denote the equation of l₁ as y = kx + c. But since it passes through M(-1, -2), I can write:-2 = k*(-1) + c => c = -2 + kSo, the equation is y = kx + (k - 2)But since the line intersects the negative semi-axes, the x-intercept and y-intercept must be negative. Let me find the intercepts.To find the x-intercept, set y = 0:0 = kx + (k - 2)=> kx = -(k - 2)=> x = -(k - 2)/k = (2 - k)/kSimilarly, the y-intercept is when x = 0:y = 0 + (k - 2) = k - 2But since both intercepts must be negative, let's see what that implies for k.For the x-intercept (2 - k)/k < 0:Case 1: (2 - k) > 0 and k < 0=> 2 - k > 0 => k < 2But k < 0, so overall k < 0.Case 2: (2 - k) < 0 and k > 0=> 2 - k < 0 => k > 2But k > 0, so overall k > 2.But the y-intercept is k - 2. For the y-intercept to be negative, k - 2 < 0 => k < 2.So, combining both conditions:From x-intercept, k < 0 or k > 2From y-intercept, k < 2Thus, the only possible case is k < 0.So, the slope k must be negative.Therefore, the equation of line l₁ is y = kx + (k - 2), where k < 0.Now, the triangle formed with the negative semi-axes will have vertices at (0,0), ((2 - k)/k, 0), and (0, k - 2). The area of this triangle is (1/2)*|x-intercept|*|y-intercept|.So, let's compute the area S:S = (1/2)*| (2 - k)/k | * |k - 2|But since k < 0, let's analyze the absolute values.First, (2 - k)/k: since k < 0, denominator is negative. Numerator: 2 - k. Since k is negative, -k is positive, so 2 - k = 2 + |k|, which is positive. So, (2 - k)/k is negative. Hence, |(2 - k)/k| = -(2 - k)/k = (k - 2)/kSimilarly, |k - 2|: since k < 0, k - 2 is negative, so |k - 2| = -(k - 2) = 2 - kTherefore, S = (1/2)*[(k - 2)/k]*(2 - k)Simplify:(k - 2)/k * (2 - k) = (k - 2)(2 - k)/k = -(2 - k)^2 / kSo, S = (1/2)*[-(2 - k)^2 / k] = (1/2)*( (2 - k)^2 / (-k) )But since k < 0, -k > 0, so S is positive.Let me rewrite S:S = (1/2)*( (2 - k)^2 / (-k) )Let me make a substitution to simplify. Let t = -k, which is positive because k < 0.So, t = -k => k = -tThen, 2 - k = 2 - (-t) = 2 + tThus, S becomes:S = (1/2)*( (2 + t)^2 / t )So, S(t) = (1/2)*( (t^2 + 4t + 4)/t ) = (1/2)*( t + 4 + 4/t )So, S(t) = (1/2)*( t + 4 + 4/t )Now, to minimize S(t), we can consider the function f(t) = t + 4 + 4/t, t > 0.We need to find the minimum of f(t). Let's take derivative:f'(t) = 1 - 4/t²Set derivative equal to zero:1 - 4/t² = 0 => 1 = 4/t² => t² = 4 => t = 2 (since t > 0)So, the minimum occurs at t = 2.Thus, the minimal area is S = (1/2)*(2 + 4 + 4/2) = (1/2)*(2 + 4 + 2) = (1/2)*8 = 4So, the minimal area is 4.Now, we need to find the equation of line l₁ when t = 2.Recall that t = -k, so k = -t = -2.Thus, the slope k is -2.So, the equation of l₁ is y = -2x + (k - 2) = -2x + (-2 - 2) = -2x - 4Wait, let me check:Earlier, we had c = k - 2, so with k = -2, c = -2 - 2 = -4.So, the equation is y = -2x - 4But let me write it in standard form:y = -2x - 4Bring all terms to left:2x + y + 4 = 0So, the equation is 2x + y + 4 = 0Let me verify if this line passes through M(-1, -2):2*(-1) + (-2) + 4 = -2 -2 +4 = 0. Yes, it does.Also, let's check the intercepts:x-intercept: set y=0: 2x + 0 +4=0 => 2x = -4 => x = -2y-intercept: set x=0: 0 + y +4=0 => y = -4So, the triangle has vertices at (0,0), (-2,0), and (0,-4). The area is (1/2)*2*4=4, which matches our earlier calculation.Therefore, the equation of l₁ is 2x + y +4 =0.Wait, but in the problem statement, it says "through point M draw line l₁ such that it forms a triangle with the negative semi-axes of the two coordinate axes, having the smallest possible area." So, I think we did it correctly.But just to make sure, let me recap:1. Found fixed point M(-1, -2)2. Expressed line l₁ through M with slope k <03. Expressed intercepts in terms of k4. Expressed area in terms of k, substituted t = -k5. Minimized the area function, found t=2, so k=-26. Found equation of l₁ as 2x + y +4=0Everything seems to check out. So, I think that's the answer.**Final Answer**The equation of line ( l_1 ) is boxed{2x + y + 4 = 0}.

💡Alright, so I have this problem here about arithmetic and geometric sequences. Let me try to understand what it's asking for and figure out how to approach it step by step.First, the problem says that the sum of the first n terms of an arithmetic sequence {a_n} is A_n, and the sum of the first n terms of a geometric sequence {b_n} is B_n. We're given that a_3 = b_3 and a_4 = b_4. Additionally, the ratio (A_5 - A_3)/(B_4 - B_2) is equal to 7. We need to find the value of (a_5 + a_3)/(b_5 + b_3).Okay, let's break this down. I know that for an arithmetic sequence, the nth term is given by a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. The sum of the first n terms, A_n, is given by the formula A_n = n/2 [2a_1 + (n-1)d].For a geometric sequence, the nth term is b_n = b_1 * q^(n-1), where b_1 is the first term and q is the common ratio. The sum of the first n terms, B_n, is given by B_n = b_1*(1 - q^n)/(1 - q) if q ≠ 1.Given that a_3 = b_3 and a_4 = b_4, I can write these as equations:1. a_3 = a_1 + 2d = b_3 = b_1*q^22. a_4 = a_1 + 3d = b_4 = b_1*q^3So, from these, I can express b_1 and q in terms of a_1 and d.From equation 1: b_1 = (a_1 + 2d)/q^2From equation 2: a_1 + 3d = b_1*q^3Substituting b_1 from equation 1 into equation 2:a_1 + 3d = [(a_1 + 2d)/q^2] * q^3 = (a_1 + 2d)*qSo, we get:a_1 + 3d = (a_1 + 2d)*qLet me solve for q:q = (a_1 + 3d)/(a_1 + 2d)Alright, so q is expressed in terms of a_1 and d. Now, let's move on to the other part of the problem.We have (A_5 - A_3)/(B_4 - B_2) = 7.First, let's compute A_5 - A_3.Using the formula for A_n:A_5 = 5/2 [2a_1 + 4d] = 5/2 [2a_1 + 4d] = 5(a_1 + 2d)A_3 = 3/2 [2a_1 + 2d] = 3/2 [2(a_1 + d)] = 3(a_1 + d)So, A_5 - A_3 = 5(a_1 + 2d) - 3(a_1 + d) = 5a_1 + 10d - 3a_1 - 3d = 2a_1 + 7dOkay, so A_5 - A_3 = 2a_1 + 7d.Now, let's compute B_4 - B_2.Using the formula for B_n:B_4 = b_1*(1 - q^4)/(1 - q)B_2 = b_1*(1 - q^2)/(1 - q)So, B_4 - B_2 = [b_1*(1 - q^4) - b_1*(1 - q^2)] / (1 - q) = b_1*(1 - q^4 - 1 + q^2)/(1 - q) = b_1*(q^2 - q^4)/(1 - q)Factor out q^2 from the numerator:= b_1*q^2*(1 - q^2)/(1 - q)Note that (1 - q^2) = (1 - q)(1 + q), so:= b_1*q^2*(1 - q)(1 + q)/(1 - q) = b_1*q^2*(1 + q)So, B_4 - B_2 = b_1*q^2*(1 + q)Therefore, the ratio (A_5 - A_3)/(B_4 - B_2) becomes:(2a_1 + 7d)/(b_1*q^2*(1 + q)) = 7But from earlier, we have b_1 = (a_1 + 2d)/q^2. Let's substitute that into the equation:(2a_1 + 7d)/[( (a_1 + 2d)/q^2 )*q^2*(1 + q)] = 7Simplify the denominator:( (a_1 + 2d)/q^2 )*q^2*(1 + q) = (a_1 + 2d)*(1 + q)So, the equation becomes:(2a_1 + 7d)/( (a_1 + 2d)*(1 + q) ) = 7Now, let's substitute q from earlier: q = (a_1 + 3d)/(a_1 + 2d)So, 1 + q = 1 + (a_1 + 3d)/(a_1 + 2d) = [ (a_1 + 2d) + (a_1 + 3d) ] / (a_1 + 2d) = (2a_1 + 5d)/(a_1 + 2d)Therefore, the equation becomes:(2a_1 + 7d)/[ (a_1 + 2d)*(2a_1 + 5d)/(a_1 + 2d) ) ] = 7Simplify the denominator:(a_1 + 2d)*(2a_1 + 5d)/(a_1 + 2d) = 2a_1 + 5dSo, we have:(2a_1 + 7d)/(2a_1 + 5d) = 7Let me write that equation:(2a_1 + 7d)/(2a_1 + 5d) = 7Cross-multiplying:2a_1 + 7d = 7*(2a_1 + 5d)Expand the right side:2a_1 + 7d = 14a_1 + 35dBring all terms to one side:2a_1 + 7d - 14a_1 - 35d = 0Combine like terms:(2a_1 - 14a_1) + (7d - 35d) = 0-12a_1 - 28d = 0Let me solve for a_1:-12a_1 = 28da_1 = (28d)/(-12) = -7d/3So, a_1 = -7d/3Alright, so we have a_1 in terms of d. Now, let's find q.Earlier, we had q = (a_1 + 3d)/(a_1 + 2d)Substitute a_1 = -7d/3:q = ( (-7d/3) + 3d ) / ( (-7d/3) + 2d )Simplify numerator and denominator:Numerator: (-7d/3 + 9d/3) = (2d/3)Denominator: (-7d/3 + 6d/3) = (-d/3)So, q = (2d/3)/(-d/3) = (2d/3)*(-3/d) = -2So, q = -2Alright, so now we know that a_1 = -7d/3 and q = -2.Now, let's find the required ratio: (a_5 + a_3)/(b_5 + b_3)First, compute a_5 and a_3.a_5 = a_1 + 4d = (-7d/3) + 4d = (-7d/3 + 12d/3) = 5d/3a_3 = a_1 + 2d = (-7d/3) + 2d = (-7d/3 + 6d/3) = (-d)/3So, a_5 + a_3 = (5d/3) + (-d/3) = (4d)/3Now, compute b_5 and b_3.First, recall that b_1 = (a_1 + 2d)/q^2We have a_1 = -7d/3, so a_1 + 2d = (-7d/3 + 6d/3) = (-d)/3q = -2, so q^2 = 4Therefore, b_1 = (-d/3)/4 = (-d)/12So, b_1 = -d/12Now, compute b_5 and b_3.b_5 = b_1*q^4 = (-d/12)*(-2)^4 = (-d/12)*(16) = (-16d)/12 = (-4d)/3b_3 = b_1*q^2 = (-d/12)*(4) = (-4d)/12 = (-d)/3So, b_5 + b_3 = (-4d/3) + (-d/3) = (-5d)/3Therefore, the ratio (a_5 + a_3)/(b_5 + b_3) is:(4d/3)/(-5d/3) = (4d/3)*(-3/(5d)) = (4*(-3))/(3*5) = (-12)/15 = -4/5So, the value is -4/5.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, a_1 = -7d/3, correct.q = -2, correct.a_5 = a_1 + 4d = (-7d/3) + 4d = (-7d + 12d)/3 = 5d/3, correct.a_3 = a_1 + 2d = (-7d/3) + 2d = (-7d + 6d)/3 = (-d)/3, correct.So, a_5 + a_3 = 5d/3 - d/3 = 4d/3, correct.b_1 = (a_1 + 2d)/q^2 = (-d/3)/4 = -d/12, correct.b_5 = b_1*q^4 = (-d/12)*(16) = -16d/12 = -4d/3, correct.b_3 = b_1*q^2 = (-d/12)*(4) = -4d/12 = -d/3, correct.So, b_5 + b_3 = -4d/3 - d/3 = -5d/3, correct.Thus, the ratio is (4d/3)/(-5d/3) = -4/5, correct.Okay, so I think my calculations are correct. The final answer is -4/5.

💡Okay, so I have this problem where I'm given that (x) is a real number and (x^3 + frac{1}{x^3} = 110). I need to find the value of (x + frac{1}{x}). Hmm, I remember that there are some algebraic identities that relate expressions like (x + frac{1}{x}) to higher powers like (x^3 + frac{1}{x^3}). Maybe I can use one of those identities to solve this problem.Let me think. I recall that ((x + frac{1}{x})^3) expands to something involving (x^3) and (frac{1}{x^3}). Let me try expanding that to see if it helps.So, let's let (a = x + frac{1}{x}). Then, (a^3) would be:[a^3 = left(x + frac{1}{x}right)^3]Expanding this, I get:[a^3 = x^3 + 3x^2 cdot frac{1}{x} + 3x cdot frac{1}{x^2} + frac{1}{x^3}]Simplifying each term:- The first term is (x^3).- The second term is (3x^2 cdot frac{1}{x} = 3x).- The third term is (3x cdot frac{1}{x^2} = frac{3}{x}).- The fourth term is (frac{1}{x^3}).So, putting it all together:[a^3 = x^3 + frac{1}{x^3} + 3x + frac{3}{x}]Wait, I can factor out the 3 from the middle terms:[a^3 = x^3 + frac{1}{x^3} + 3left(x + frac{1}{x}right)]But I know that (x + frac{1}{x} = a), so substituting that in:[a^3 = x^3 + frac{1}{x^3} + 3a]Now, the problem gives me that (x^3 + frac{1}{x^3} = 110). Let me substitute that into the equation:[a^3 = 110 + 3a]So, now I have the equation:[a^3 - 3a - 110 = 0]This is a cubic equation in terms of (a). I need to solve for (a). Since (a) is a real number (because (x) is real and (x + frac{1}{x}) is also real), I can try to find real roots of this cubic equation.I remember that for cubic equations, one way to find roots is to use the Rational Root Theorem, which suggests that any rational root, expressed as a fraction (frac{p}{q}), has (p) as a factor of the constant term and (q) as a factor of the leading coefficient. In this case, the constant term is -110 and the leading coefficient is 1. So, the possible rational roots are the factors of 110, which are (pm1, pm2, pm5, pm10, pm11, pm22, pm55, pm110).Let me test these possible roots one by one.Starting with (a = 1):[1^3 - 3(1) - 110 = 1 - 3 - 110 = -112 neq 0]Not a root.Next, (a = -1):[(-1)^3 - 3(-1) - 110 = -1 + 3 - 110 = -108 neq 0]Not a root.Trying (a = 2):[2^3 - 3(2) - 110 = 8 - 6 - 110 = -108 neq 0]Not a root.(a = -2):[(-2)^3 - 3(-2) - 110 = -8 + 6 - 110 = -112 neq 0]Not a root.Moving on to (a = 5):[5^3 - 3(5) - 110 = 125 - 15 - 110 = 0]Oh, that works! So, (a = 5) is a root.To confirm, let's factor the cubic equation using this root. If (a = 5) is a root, then ((a - 5)) is a factor. Let's perform polynomial division or use synthetic division to factor it out.Using synthetic division:Divide (a^3 - 3a - 110) by (a - 5).Set up the coefficients: 1 (for (a^3)), 0 (for (a^2)), -3 (for (a)), -110.Bring down the 1.Multiply 1 by 5 (the root), get 5. Add to the next coefficient: 0 + 5 = 5.Multiply 5 by 5, get 25. Add to the next coefficient: -3 + 25 = 22.Multiply 22 by 5, get 110. Add to the last coefficient: -110 + 110 = 0. Perfect, no remainder.So, the cubic factors as:[(a - 5)(a^2 + 5a + 22) = 0]Now, set each factor equal to zero:1. (a - 5 = 0) gives (a = 5).2. (a^2 + 5a + 22 = 0). Let's find the discriminant to see if there are real roots:Discriminant (D = 5^2 - 4 cdot 1 cdot 22 = 25 - 88 = -63).Since the discriminant is negative, there are no real roots from this quadratic factor. Therefore, the only real solution is (a = 5).So, (x + frac{1}{x} = 5).Wait, just to make sure I didn't make any mistakes. Let me verify the steps again.1. Let (a = x + frac{1}{x}).2. Expanded (a^3) correctly to get (x^3 + frac{1}{x^3} + 3a).3. Substituted (x^3 + frac{1}{x^3} = 110) to get (a^3 = 110 + 3a).4. Formed the cubic equation (a^3 - 3a - 110 = 0).5. Applied the Rational Root Theorem, found (a = 5) as a root.6. Factored the cubic and confirmed that the quadratic has no real roots, so (a = 5) is the only real solution.Everything seems to check out. I think I did it right.**Final Answer**The value of (x + frac{1}{x}) is boxed{5}.