💡First, I need to determine the number of handshakes that occur when 12 people each shake hands with every other person exactly once. This is a combination problem where the order of interaction does not matter. The formula for combinations is:[text{Number of handshakes} = binom{n}{2} = frac{n(n-1)}{2}]Plugging in 12 for ( n ):[text{Number of handshakes} = frac{12 times 11}{2} = 66]Next, I need to calculate the number of high-fives. Since each person also high-fives every other person exactly once, the calculation is identical to the handshakes:[text{Number of high-fives} = binom{12}{2} = 66]Therefore, there are 66 handshakes and 66 high-fives at the gathering.

💡First, identify the total number of friends Mrs. Hilt has, which is 15.Next, determine the number of friends who cannot attend the movies, which is 7.To find out how many friends she can take to the movies, subtract the number of friends who can't go from the total number of friends.So, 15 minus 7 equals 8.Therefore, Mrs. Hilt can take 8 friends to the movies.

💡Okay, so I have this problem about a right circular cone with a vertex angle of 90 degrees. The diameter of the base is 24 inches, and I need to find the volume of a sphere that's placed inside the cone, touching the sides and resting on the table. Hmm, let me try to visualize this.First, the cone is pointing upwards, and when they say the cross-section has a vertex angle of 90 degrees, that means if I slice the cone vertically through its axis, I get an isosceles right triangle. The base of this triangle is the diameter of the cone's base, which is 24 inches. So, the radius of the cone's base is half of that, which is 12 inches.Since the cross-section is a right isosceles triangle, the two legs are equal. In this case, the legs would be the height of the cone and the radius. Wait, no, actually, in a right isosceles triangle, the legs are equal, and the hypotenuse is longer. So, if the base is 24 inches, that's the hypotenuse of the triangle, right? Because the vertex angle is 90 degrees, so the two sides from the vertex to the base are equal.Let me draw this in my mind. The cone has a base diameter of 24 inches, so the base radius is 12 inches. The cross-section is a right isosceles triangle with legs equal to the height of the cone and the radius. Wait, no, that can't be because the base is 24 inches, which is the hypotenuse. So, if the triangle is right-angled at the vertex, then the two sides from the vertex to the base are equal, each being the radius and the height.Wait, maybe I need to clarify. If the cross-section is a right triangle with a vertex angle of 90 degrees, then the two sides meeting at the vertex are equal. So, the height of the cone and the radius of the base are equal? But the diameter is 24 inches, so the radius is 12 inches. That would mean the height is also 12 inches? But that seems too short for a cone with a base diameter of 24 inches. Maybe I'm misunderstanding.Wait, no, in a right isosceles triangle, the legs are equal, and the hypotenuse is leg * sqrt(2). So, if the base of the cone is 24 inches, that's the hypotenuse of the cross-sectional triangle. So, each leg (the height and the radius) would be 24 / sqrt(2) = 12*sqrt(2) inches. Wait, that doesn't make sense because the radius is given as 12 inches. Hmm, maybe I need to think differently.Let me try to break it down step by step. The cone has a base diameter of 24 inches, so the radius is 12 inches. The cross-section is a right triangle with a vertex angle of 90 degrees. So, the two sides from the vertex to the base are equal. Let me denote the height of the cone as h and the radius as r = 12 inches.In the right triangle, the two legs are h and r, and the hypotenuse is the slant height. But since the vertex angle is 90 degrees, the triangle is right-angled at the vertex, so the legs are h and r, and the hypotenuse is the slant height. Wait, but in a right isosceles triangle, the legs are equal, so h must equal r? But r is 12 inches, so h would also be 12 inches. But then the slant height would be 12*sqrt(2) inches.Wait, that seems possible. So, the height of the cone is 12 inches, and the radius is 12 inches. That makes the slant height 12*sqrt(2) inches. Okay, that makes sense.Now, we need to place a sphere inside this cone such that it's tangent to the sides and rests on the table. So, the sphere touches the cone along its sides and sits on the base of the cone. We need to find the volume of this sphere.To find the volume, we need the radius of the sphere. Let's denote the radius of the sphere as R. The sphere is tangent to the cone's sides and rests on the base, so the center of the sphere is R inches above the base. So, the center is at a height R from the table.Now, let's consider the cross-sectional view. The cone is a right isosceles triangle with legs of 12 inches each. The sphere inscribed in this triangle will touch both legs and the base. The center of the sphere is at (R, R) in a coordinate system where the vertex of the cone is at (0,0), and the base is along the x-axis from (0,0) to (24,0). Wait, no, actually, in the cross-section, the cone is a triangle with vertex at (0,0), base along the x-axis from (-12,0) to (12,0), and the apex at (0,12). Hmm, maybe I should set it up that way.Alternatively, let's set the vertex at (0,0), and the base of the cone is along the x-axis from (-12,0) to (12,0), and the apex is at (0,12). Wait, no, because the height is 12 inches, so the apex is at (0,12). The sphere is sitting on the base, so its center is at (0, R), and it touches the sides of the cone.The sides of the cone are the lines from (0,12) to (12,0) and from (0,12) to (-12,0). Let's find the equation of one of these sides, say the right side from (0,12) to (12,0). The slope of this line is (0 - 12)/(12 - 0) = -1. So, the equation is y = -x + 12.The sphere is centered at (0, R) with radius R. The distance from the center of the sphere to the side of the cone must be equal to the radius R. The distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a^2 + b^2). So, let's write the equation of the side as x + y - 12 = 0. Then, the distance from (0, R) to this line is |0 + R - 12| / sqrt(1 + 1) = |R - 12| / sqrt(2).This distance must equal the radius R, so we have |R - 12| / sqrt(2) = R. Since R is positive and less than 12 (because the sphere is inside the cone), we can drop the absolute value: (12 - R) / sqrt(2) = R.Now, let's solve for R:(12 - R) / sqrt(2) = RMultiply both sides by sqrt(2):12 - R = R * sqrt(2)Bring R terms to one side:12 = R * sqrt(2) + RFactor out R:12 = R (sqrt(2) + 1)Therefore, R = 12 / (sqrt(2) + 1)To rationalize the denominator, multiply numerator and denominator by (sqrt(2) - 1):R = 12 (sqrt(2) - 1) / [(sqrt(2) + 1)(sqrt(2) - 1)] = 12 (sqrt(2) - 1) / (2 - 1) = 12 (sqrt(2) - 1)So, R = 12 (sqrt(2) - 1) inches.Now, the volume of the sphere is (4/3)πR³.Let's compute R³:R = 12 (sqrt(2) - 1)R³ = [12 (sqrt(2) - 1)]³ = 12³ (sqrt(2) - 1)³ = 1728 (sqrt(2) - 1)³Let's compute (sqrt(2) - 1)³:Let’s denote a = sqrt(2), so (a - 1)³ = a³ - 3a² + 3a - 1.Compute each term:a³ = (sqrt(2))³ = 2 * sqrt(2)3a² = 3*(2) = 63a = 3*sqrt(2)So, (a - 1)³ = 2*sqrt(2) - 6 + 3*sqrt(2) - 1 = (2*sqrt(2) + 3*sqrt(2)) + (-6 - 1) = 5*sqrt(2) - 7Therefore, R³ = 1728*(5*sqrt(2) - 7)So, the volume is:V = (4/3)π * 1728*(5*sqrt(2) - 7) = (4/3)*1728*(5*sqrt(2) - 7) * πCompute (4/3)*1728:1728 / 3 = 576576 * 4 = 2304So, V = 2304*(5*sqrt(2) - 7)πWait, that seems complicated. Did I make a mistake somewhere?Wait, let me double-check the calculation of R. Earlier, I set up the equation (12 - R)/sqrt(2) = R, which led to R = 12/(sqrt(2)+1). Then, rationalizing, R = 12(sqrt(2)-1). That seems correct.But when I computed R³, I think I might have made an error in expanding (sqrt(2)-1)³. Let me recompute that.Let me compute (sqrt(2) - 1)^3 step by step.First, compute (sqrt(2) - 1)^2:= (sqrt(2))^2 - 2*sqrt(2)*1 + (1)^2= 2 - 2*sqrt(2) + 1= 3 - 2*sqrt(2)Now, multiply by (sqrt(2) - 1):= (3 - 2*sqrt(2))(sqrt(2) - 1)= 3*sqrt(2) - 3*1 - 2*sqrt(2)*sqrt(2) + 2*sqrt(2)*1= 3*sqrt(2) - 3 - 2*2 + 2*sqrt(2)= 3*sqrt(2) - 3 - 4 + 2*sqrt(2)= (3*sqrt(2) + 2*sqrt(2)) + (-3 - 4)= 5*sqrt(2) - 7Okay, that part was correct.So, R³ = 1728*(5*sqrt(2) - 7)Thus, V = (4/3)π * 1728*(5*sqrt(2) - 7) = 2304*(5*sqrt(2) - 7)πWait, but the problem says to express the answer in terms of π, so maybe that's acceptable. However, I'm wondering if there's a simpler way to express this or if I made a mistake in the setup.Alternatively, maybe I should have considered similar triangles or another approach.Let me think differently. The sphere is tangent to the cone and the base. The center of the sphere is at height R above the base. The cone has a height of 12 inches and a base radius of 12 inches. So, the cone's slope is 1 (since radius and height are equal).The sphere touches the cone's side, so the distance from the center of the sphere to the cone's side is R. The cone's side makes a 45-degree angle with the base because it's a right isosceles triangle.So, the distance from the center to the side can also be expressed in terms of the angle. The distance from a point to a line at an angle θ is R = (distance along the axis) * sin(θ). Wait, maybe not exactly, but perhaps using trigonometry.Alternatively, considering the right triangle formed by the center of the sphere, the point of tangency on the cone's side, and the projection of the center onto the cone's side.Wait, maybe it's better to use similar triangles. The cone has a height of 12 and radius 12, so the slope is 1. The sphere of radius R is inside, so from the apex to the center of the sphere is 12 - R. The line from the apex to the point of tangency on the sphere makes a right angle with the tangent line.Wait, perhaps using similar triangles: the big triangle (cone) and the smaller triangle from the apex to the center of the sphere.The big triangle has height 12 and base radius 12. The smaller triangle, from the apex to the center of the sphere, has height 12 - R and base radius equal to the distance from the center to the side, which is R / sin(45°), because the angle between the cone's side and the axis is 45 degrees.Wait, let me clarify. The cone's side makes a 45-degree angle with the axis because it's a right isosceles triangle. So, the distance from the center to the side is R / sin(45°). But also, this distance is equal to the base radius of the smaller triangle, which is (12 - R) * tan(45°). Since tan(45°) = 1, the base radius of the smaller triangle is 12 - R.Wait, that might not be correct. Let me think again.The distance from the center to the side is R, and this distance is also equal to (12 - R) * sin(45°), because in the smaller triangle, the height is 12 - R, and the angle is 45 degrees, so the opposite side (distance from center to side) is (12 - R) * sin(45°).So, R = (12 - R) * sin(45°)Since sin(45°) = sqrt(2)/2, we have:R = (12 - R) * sqrt(2)/2Multiply both sides by 2:2R = (12 - R) * sqrt(2)Expand:2R = 12*sqrt(2) - R*sqrt(2)Bring all R terms to one side:2R + R*sqrt(2) = 12*sqrt(2)Factor R:R (2 + sqrt(2)) = 12*sqrt(2)Thus, R = (12*sqrt(2)) / (2 + sqrt(2))Rationalize the denominator by multiplying numerator and denominator by (2 - sqrt(2)):R = [12*sqrt(2)*(2 - sqrt(2))] / [(2 + sqrt(2))(2 - sqrt(2))] = [12*sqrt(2)*(2 - sqrt(2))] / (4 - 2) = [12*sqrt(2)*(2 - sqrt(2))] / 2 = 6*sqrt(2)*(2 - sqrt(2)) = 6*(2*sqrt(2) - 2) = 12*sqrt(2) - 12Wait, that's different from what I got earlier. Earlier, I had R = 12(sqrt(2) - 1), which is the same as 12*sqrt(2) - 12. So, that matches. So, R = 12(sqrt(2) - 1).So, the radius of the sphere is 12(sqrt(2) - 1) inches.Therefore, the volume is (4/3)πR³ = (4/3)π[12(sqrt(2) - 1)]³.As I computed earlier, [12(sqrt(2) - 1)]³ = 1728*(5*sqrt(2) - 7). So, the volume is 2304*(5*sqrt(2) - 7)π.Wait, but that seems quite complicated. Maybe I made a mistake in expanding (sqrt(2) - 1)^3. Let me check again.(sqrt(2) - 1)^3:First, (sqrt(2) - 1)^2 = 2 - 2*sqrt(2) + 1 = 3 - 2*sqrt(2).Then, multiply by (sqrt(2) - 1):(3 - 2*sqrt(2))(sqrt(2) - 1) = 3*sqrt(2) - 3 - 2*sqrt(2)*sqrt(2) + 2*sqrt(2) = 3*sqrt(2) - 3 - 4 + 2*sqrt(2) = 5*sqrt(2) - 7.Yes, that's correct.So, R³ = 1728*(5*sqrt(2) - 7).Thus, V = (4/3)π * 1728*(5*sqrt(2) - 7) = 2304*(5*sqrt(2) - 7)π.Hmm, that seems correct, but I'm a bit surprised it's not a simpler expression. Maybe there's a different approach that leads to a simpler answer.Wait, perhaps I made a mistake in the setup. Let me try another method.Consider the cone and the inscribed sphere. The sphere touches the cone along a circle, and the center of the sphere is along the axis of the cone. Let's denote the radius of the sphere as R.The cone has a height H = 12 inches and a base radius R_cone = 12 inches. The sphere of radius R is inside, so the distance from the apex to the center of the sphere is H - R = 12 - R.The slope of the cone's side is R_cone / H = 12 / 12 = 1, so the angle θ between the axis and the side is 45 degrees.The distance from the center of the sphere to the side of the cone is R, and this distance can also be expressed as (H - R) * sin(θ). Since θ = 45 degrees, sin(45) = sqrt(2)/2.So, R = (12 - R) * sqrt(2)/2Multiply both sides by 2:2R = (12 - R) * sqrt(2)Expand:2R = 12*sqrt(2) - R*sqrt(2)Bring R terms to one side:2R + R*sqrt(2) = 12*sqrt(2)Factor R:R (2 + sqrt(2)) = 12*sqrt(2)Thus, R = (12*sqrt(2)) / (2 + sqrt(2)) = 12(sqrt(2) - 1), as before.So, the radius R is 12(sqrt(2) - 1). Therefore, the volume is (4/3)πR³.As computed earlier, R³ = 1728*(5*sqrt(2) - 7), so V = 2304*(5*sqrt(2) - 7)π.Wait, but this seems quite involved. Maybe I should check if there's a simpler way or if I made a mistake in the expansion.Alternatively, perhaps I can express the volume in terms of R without expanding R³. Since R = 12(sqrt(2) - 1), then R³ = 12³*(sqrt(2) - 1)^3 = 1728*(5*sqrt(2) - 7), as before.So, the volume is indeed 2304*(5*sqrt(2) - 7)π cubic inches.Wait, but the problem says to express the answer in terms of π, so this is acceptable. However, I'm wondering if there's a simpler form or if I made a mistake in the calculation.Alternatively, maybe I can factor out something. Let me see:2304*(5*sqrt(2) - 7)π = 2304π*(5*sqrt(2) - 7)But I don't think it simplifies further. So, perhaps that's the answer.Wait, but let me check the initial assumption about the cone's height. Earlier, I thought the height was 12 inches because the radius is 12 inches and the triangle is right isosceles. But wait, in a right isosceles triangle, the legs are equal, so if the base is 24 inches, which is the hypotenuse, then each leg is 24 / sqrt(2) = 12*sqrt(2) inches. Wait, that contradicts my earlier conclusion.Wait, hold on, I think I made a mistake here. Let me clarify.The cross-section is a right triangle with a vertex angle of 90 degrees. The base of the cone is 24 inches, which is the diameter, so the radius is 12 inches. The cross-section is a triangle with base 24 inches and vertex angle 90 degrees. Therefore, the two equal sides (legs) are the height of the cone and the radius. Wait, no, the base is 24 inches, which is the hypotenuse of the right triangle, so the legs are equal and each is 24 / sqrt(2) = 12*sqrt(2) inches.Wait, that makes more sense. Because in a right isosceles triangle, if the hypotenuse is 24, then each leg is 24 / sqrt(2) = 12*sqrt(2). So, the height of the cone is 12*sqrt(2) inches, and the radius is also 12*sqrt(2) inches. But wait, the radius is given as 12 inches because the diameter is 24 inches. So, that's a contradiction.Wait, no, the cross-section is a right triangle with vertex angle 90 degrees, and the base of the triangle is the diameter of the cone's base, which is 24 inches. Therefore, the two legs of the triangle are the height of the cone and the radius of the base. But since the triangle is right-angled at the vertex, the two legs are equal, so the height equals the radius. But the radius is 12 inches, so the height is also 12 inches. Therefore, the slant height (hypotenuse) is 12*sqrt(2) inches.Wait, that makes sense. So, the cone has a height of 12 inches and a base radius of 12 inches. Therefore, the slope of the cone's side is 1, as I initially thought.So, going back, the sphere is inside this cone with height 12 and radius 12. The sphere's radius R is found by R = 12(sqrt(2) - 1), as before.Therefore, the volume is (4/3)π[12(sqrt(2) - 1)]³ = 2304*(5*sqrt(2) - 7)π.Wait, but let me check if this makes sense. If the cone's height is 12 inches, and the sphere's radius is about 12*(1.414 - 1) = 12*(0.414) ≈ 5 inches, so the sphere is about 5 inches in radius, which seems reasonable inside a 12-inch tall cone.But let me compute the numerical value to see if it makes sense.Compute 5*sqrt(2) - 7:sqrt(2) ≈ 1.414, so 5*1.414 ≈ 7.07, so 7.07 - 7 = 0.07.So, 2304*0.07 ≈ 161.28.So, the volume is approximately 161.28π cubic inches.Wait, but if R ≈ 5 inches, then the volume should be (4/3)π*(5)^3 ≈ (4/3)π*125 ≈ 166.67π, which is close to 161.28π. Hmm, that's a bit off, but considering the approximation, it's close enough.Alternatively, maybe I made a mistake in the exact calculation. Let me compute 5*sqrt(2) - 7 more accurately.sqrt(2) ≈ 1.414213565*sqrt(2) ≈ 7.07106787.0710678 - 7 = 0.0710678So, 2304*0.0710678 ≈ 2304*0.0710678 ≈ 163.68πWhich is closer to the approximate 166.67π, considering that R is approximately 5 inches, but actually R = 12(sqrt(2)-1) ≈ 12*(0.4142) ≈ 4.97 inches, so R³ ≈ 123.33, and (4/3)π*123.33 ≈ 164.44π, which is close to 163.68π. So, that seems consistent.Therefore, I think my calculation is correct, and the volume is 2304*(5*sqrt(2) - 7)π cubic inches.Wait, but I'm a bit concerned because the problem seems to expect a simpler answer, maybe just a multiple of π without the radicals. Did I make a mistake in the setup?Wait, let me think again. Maybe I should have considered the cone's height differently. If the cross-section is a right isosceles triangle with base 24 inches, then the legs are each 24 / sqrt(2) = 12*sqrt(2) inches. So, the height of the cone is 12*sqrt(2) inches, and the radius is also 12*sqrt(2) inches. Wait, but the diameter is 24 inches, so the radius is 12 inches, not 12*sqrt(2). So, that's a contradiction.Wait, no, the cross-section is a right triangle with base 24 inches and vertex angle 90 degrees. Therefore, the two legs are equal, each being the height and the radius. But the radius is 12 inches, so the height must also be 12 inches, making the hypotenuse (slant height) 12*sqrt(2) inches. So, that's correct.Therefore, the cone has height 12 inches and radius 12 inches. So, the slope of the cone's side is 1, as I initially thought.Thus, the sphere's radius R is 12(sqrt(2) - 1), and the volume is 2304*(5*sqrt(2) - 7)π.Wait, but maybe I can factor out something else. Let me see:2304*(5*sqrt(2) - 7) = 2304*5*sqrt(2) - 2304*7 = 11520*sqrt(2) - 16128.But that doesn't seem helpful.Alternatively, maybe I can write it as 2304π*(5*sqrt(2) - 7).Yes, that's the simplest form.Therefore, the volume of the sphere is 2304π*(5*sqrt(2) - 7) cubic inches.Wait, but I'm still a bit unsure because the problem seems to expect a simpler answer. Maybe I made a mistake in the calculation of R³.Let me recompute R³:R = 12(sqrt(2) - 1)R³ = [12(sqrt(2) - 1)]³ = 12³*(sqrt(2) - 1)³ = 1728*(5*sqrt(2) - 7)Yes, that's correct.So, the volume is (4/3)π*1728*(5*sqrt(2) - 7) = 2304π*(5*sqrt(2) - 7).I think that's the correct answer, even though it's a bit complex.

💡First, identify the total number of fowls Jose has, which is 46.Next, note that Jose has 18 ducks.To find the number of chickens, subtract the number of ducks from the total number of fowls.So, 46 minus 18 equals 28.Therefore, Jose has 28 chickens.

💡Alright, so I have this problem here where I need to find the equation of a line ( l ) that passes through the point ( M(1, 0) ) and intersects the circle ( x^2 + y^2 = 5 ) at points ( A ) and ( B ). Point ( A ) is in the first quadrant, and there's this vector condition: ( overrightarrow{BM} = 2overrightarrow{MA} ). Hmm, okay, let me try to unpack this step by step.First, I know that the line passes through ( M(1, 0) ), so I can represent the line in a parametric form or maybe in slope-intercept form. Since it's a line in the Cartesian plane, maybe using the slope-intercept form would be easier. But wait, I don't know the slope yet. Alternatively, I can let the equation of the line be ( y = m(x - 1) ), where ( m ) is the slope. That might be a good start because it ensures the line passes through ( (1, 0) ).So, let me write that down: ( y = m(x - 1) ). Now, this line intersects the circle ( x^2 + y^2 = 5 ). To find the points of intersection, I can substitute ( y ) from the line equation into the circle equation. Let's do that:Substitute ( y = m(x - 1) ) into ( x^2 + y^2 = 5 ):( x^2 + [m(x - 1)]^2 = 5 )Expanding that:( x^2 + m^2(x^2 - 2x + 1) = 5 )Let me distribute the ( m^2 ):( x^2 + m^2x^2 - 2m^2x + m^2 = 5 )Now, combine like terms:( (1 + m^2)x^2 - 2m^2x + (m^2 - 5) = 0 )So, that's a quadratic equation in terms of ( x ). Let me denote this as:( (1 + m^2)x^2 - 2m^2x + (m^2 - 5) = 0 )Let me call this Equation (1).Now, this quadratic equation will have two solutions corresponding to points ( A ) and ( B ). Let me denote the coordinates of ( A ) as ( (x_1, y_1) ) and ( B ) as ( (x_2, y_2) ). Since ( A ) is in the first quadrant, both ( x_1 ) and ( y_1 ) are positive.Given the vector condition ( overrightarrow{BM} = 2overrightarrow{MA} ), I need to translate this into mathematical terms. Let me recall that vectors can be represented as differences in coordinates. So, ( overrightarrow{BM} = M - B ) and ( overrightarrow{MA} = A - M ). The condition says that ( overrightarrow{BM} = 2overrightarrow{MA} ), so:( M - B = 2(A - M) )Let me write this out in coordinates. ( M ) is ( (1, 0) ), ( B ) is ( (x_2, y_2) ), and ( A ) is ( (x_1, y_1) ). So:( (1 - x_2, 0 - y_2) = 2(x_1 - 1, y_1 - 0) )Which simplifies to:( (1 - x_2, -y_2) = (2x_1 - 2, 2y_1) )Therefore, we can set up the following system of equations:1. ( 1 - x_2 = 2x_1 - 2 )2. ( -y_2 = 2y_1 )Let me solve these equations.From equation 1:( 1 - x_2 = 2x_1 - 2 )Let me rearrange:( -x_2 = 2x_1 - 3 )Multiply both sides by -1:( x_2 = -2x_1 + 3 )From equation 2:( -y_2 = 2y_1 )Multiply both sides by -1:( y_2 = -2y_1 )So, we have expressions for ( x_2 ) and ( y_2 ) in terms of ( x_1 ) and ( y_1 ). That's useful because we can now relate the coordinates of points ( A ) and ( B ).Now, going back to Equation (1), which is the quadratic equation in ( x ). The solutions to this equation are ( x_1 ) and ( x_2 ). So, using Vieta's formulas, we know that:Sum of roots: ( x_1 + x_2 = frac{2m^2}{1 + m^2} )Product of roots: ( x_1 x_2 = frac{m^2 - 5}{1 + m^2} )Similarly, since ( y = m(x - 1) ), we can express ( y_1 ) and ( y_2 ) in terms of ( x_1 ) and ( x_2 ):( y_1 = m(x_1 - 1) )( y_2 = m(x_2 - 1) )But from equation 2, we have ( y_2 = -2y_1 ). Let me substitute ( y_2 ):( m(x_2 - 1) = -2m(x_1 - 1) )Assuming ( m neq 0 ) (since if ( m = 0 ), the line would be horizontal, and I don't think that would satisfy the vector condition given), we can divide both sides by ( m ):( x_2 - 1 = -2(x_1 - 1) )Simplify:( x_2 - 1 = -2x_1 + 2 )Bring all terms to one side:( x_2 + 2x_1 = 3 )Wait, that's the same as equation 1. So, that's consistent. So, we have ( x_2 = -2x_1 + 3 ), as before.So, now, let's use Vieta's formulas. We have ( x_1 + x_2 = frac{2m^2}{1 + m^2} ). But since ( x_2 = -2x_1 + 3 ), substitute that into the sum:( x_1 + (-2x_1 + 3) = frac{2m^2}{1 + m^2} )Simplify:( -x_1 + 3 = frac{2m^2}{1 + m^2} )Let me write this as:( -x_1 = frac{2m^2}{1 + m^2} - 3 )Multiply both sides by -1:( x_1 = 3 - frac{2m^2}{1 + m^2} )Combine the terms:( x_1 = frac{3(1 + m^2) - 2m^2}{1 + m^2} )Simplify numerator:( 3 + 3m^2 - 2m^2 = 3 + m^2 )So,( x_1 = frac{3 + m^2}{1 + m^2} )Okay, so that's ( x_1 ) in terms of ( m ). Now, let's find ( x_2 ) using ( x_2 = -2x_1 + 3 ):( x_2 = -2left( frac{3 + m^2}{1 + m^2} right) + 3 )Simplify:( x_2 = frac{-6 - 2m^2}{1 + m^2} + 3 )Express 3 as ( frac{3(1 + m^2)}{1 + m^2} ):( x_2 = frac{-6 - 2m^2 + 3 + 3m^2}{1 + m^2} )Combine like terms:( x_2 = frac{(-6 + 3) + (-2m^2 + 3m^2)}{1 + m^2} )Simplify:( x_2 = frac{-3 + m^2}{1 + m^2} )So, ( x_2 = frac{m^2 - 3}{1 + m^2} )Now, let's compute the product of roots from Vieta's formula:( x_1 x_2 = frac{m^2 - 5}{1 + m^2} )But we can also compute ( x_1 x_2 ) using the expressions we found for ( x_1 ) and ( x_2 ):( x_1 x_2 = left( frac{3 + m^2}{1 + m^2} right) left( frac{m^2 - 3}{1 + m^2} right) )Multiply the numerators:( (3 + m^2)(m^2 - 3) = m^4 - 9 )So,( x_1 x_2 = frac{m^4 - 9}{(1 + m^2)^2} )But from Vieta's formula, ( x_1 x_2 = frac{m^2 - 5}{1 + m^2} ). Therefore, set them equal:( frac{m^4 - 9}{(1 + m^2)^2} = frac{m^2 - 5}{1 + m^2} )Multiply both sides by ( (1 + m^2)^2 ):( m^4 - 9 = (m^2 - 5)(1 + m^2) )Expand the right-hand side:( (m^2 - 5)(1 + m^2) = m^4 - 5m^2 + m^2 - 5 = m^4 - 4m^2 - 5 )So, the equation becomes:( m^4 - 9 = m^4 - 4m^2 - 5 )Subtract ( m^4 ) from both sides:( -9 = -4m^2 - 5 )Add 5 to both sides:( -4 = -4m^2 )Divide both sides by -4:( 1 = m^2 )So, ( m^2 = 1 ), which implies ( m = pm 1 )Now, we need to determine which value of ( m ) is correct. Remember that point ( A ) is in the first quadrant, so ( x_1 > 0 ) and ( y_1 > 0 ). Let's check both possibilities.Case 1: ( m = 1 )Compute ( x_1 = frac{3 + (1)^2}{1 + (1)^2} = frac{4}{2} = 2 )Compute ( y_1 = m(x_1 - 1) = 1(2 - 1) = 1 )So, point ( A ) is ( (2, 1) ), which is indeed in the first quadrant.Compute ( x_2 = frac{(1)^2 - 3}{1 + (1)^2} = frac{-2}{2} = -1 )Compute ( y_2 = m(x_2 - 1) = 1(-1 - 1) = -2 )So, point ( B ) is ( (-1, -2) ). Let's verify the vector condition.Compute ( overrightarrow{BM} ) and ( overrightarrow{MA} ):( overrightarrow{BM} = M - B = (1 - (-1), 0 - (-2)) = (2, 2) )( overrightarrow{MA} = A - M = (2 - 1, 1 - 0) = (1, 1) )Check if ( overrightarrow{BM} = 2overrightarrow{MA} ):( 2overrightarrow{MA} = 2(1, 1) = (2, 2) ), which is equal to ( overrightarrow{BM} ). So, this works.Case 2: ( m = -1 )Compute ( x_1 = frac{3 + (-1)^2}{1 + (-1)^2} = frac{4}{2} = 2 )Compute ( y_1 = m(x_1 - 1) = -1(2 - 1) = -1 )So, point ( A ) is ( (2, -1) ). But this is in the fourth quadrant, not the first quadrant. Therefore, this case is invalid because ( A ) must be in the first quadrant.Thus, ( m = 1 ) is the correct slope.Therefore, the equation of line ( l ) is:( y = 1(x - 1) )Simplify:( y = x - 1 )Or, rearranged:( x - y - 1 = 0 )So, the equation of line ( l ) is ( x - y - 1 = 0 ).**Final Answer**The equation of line ( l ) is boxed{x - y - 1 = 0}.

💡To determine the smallest number of homework assignments Liam needs to complete to earn exactly 40 homework points, we analyze the structure of the homework regimen:1. **First 7 Points:** - **Assignments per Point:** 3 - **Total Assignments:** (7 times 3 = 21)2. **Next 7 Points (Points 8 to 14):** - **Assignments per Point:** 4 - **Total Assignments:** (7 times 4 = 28)3. **Third Set of 7 Points (Points 15 to 21):** - **Assignments per Point:** 5 - **Total Assignments:** (7 times 5 = 35)4. **Fourth Set of 7 Points (Points 22 to 28):** - **Assignments per Point:** 6 - **Total Assignments:** (7 times 6 = 42)5. **Fifth Set of 7 Points (Points 29 to 35):** - **Assignments per Point:** 7 - **Total Assignments:** (7 times 7 = 49)6. **Remaining 5 Points (Points 36 to 40):** - **Assignments per Point:** 8 - **Total Assignments:** (5 times 8 = 40)**Summing Up All Assignments:**[21 + 28 + 35 + 42 + 49 + 40 = 215]Therefore, the smallest number of homework assignments Liam needs to complete to earn exactly 40 homework points is **215**.