💡Okay, so I have this problem about a hyperbola, and I need to find the minimum value of |AF₂| + |BF₂| where points A and B are on the left branch of the hyperbola, and the line passing through F₁ intersects the hyperbola at A and B. Hmm, let me try to break this down step by step.First, the equation of the hyperbola is given as (x²)/9 - (y²)/6 = 1. From this, I can see that it's a standard hyperbola centered at the origin, opening to the left and right. The general form of a hyperbola is (x²/a²) - (y²/b²) = 1, so here, a² = 9 and b² = 6. That means a = 3 and b = √6.Now, for hyperbolas, the distance to the foci from the center is given by c, where c² = a² + b². Let me calculate that. So, c² = 9 + 6 = 15, which means c = √15. Therefore, the foci are located at (-√15, 0) and (√15, 0). Let me denote F₁ as the left focus (-√15, 0) and F₂ as the right focus (√15, 0).The problem states that line l passes through F₁ and intersects the left branch of the hyperbola at points A and B. So, line l is a secant line passing through F₁ and cutting the hyperbola at two points on the left branch.I need to find the minimum value of |AF₂| + |BF₂|. Hmm, okay. Let me recall some properties of hyperbolas. For any point on the hyperbola, the difference of distances to the two foci is constant and equal to 2a. So, for any point P on the hyperbola, |PF₂| - |PF₁| = 2a = 6, since a = 3.Wait, so for both points A and B, we have |AF₂| - |AF₁| = 6 and |BF₂| - |BF₁| = 6. If I add these two equations together, I get |AF₂| + |BF₂| - (|AF₁| + |BF₁|) = 12.Hmm, so |AF₂| + |BF₂| = |AF₁| + |BF₁| + 12. Now, since points A and B lie on line l, which passes through F₁, the sum |AF₁| + |BF₁| is equal to the length of segment AB. Because if you have a line passing through F₁ and intersecting the hyperbola at A and B, then F₁ is between A and B, so |AF₁| + |BF₁| = |AB|.Therefore, |AF₂| + |BF₂| = |AB| + 12. So, to minimize |AF₂| + |BF₂|, I need to minimize |AB|. So, the problem reduces to finding the minimum length of the chord AB on the left branch of the hyperbola that passes through F₁.Okay, so what is the minimum length of such a chord? I remember that for conic sections, the shortest chord through a focus is the latus rectum. Wait, but the latus rectum is the chord parallel to the directrix and passing through the focus. Is that the case here?Wait, no, the latus rectum is a specific chord, but in this case, we are looking for the minimum chord through F₁. Maybe the minor axis? But the minor axis is for an ellipse, not a hyperbola. Hmm, hyperbola doesn't have a minor axis in the same sense.Wait, perhaps the shortest chord through F₁ is the one perpendicular to the transverse axis. Let me think. The transverse axis is the x-axis in this case, since the hyperbola opens left and right. So, a line perpendicular to the x-axis would be a vertical line. But the line l passes through F₁, which is (-√15, 0). So, a vertical line through F₁ would be x = -√15. Let me see if this line intersects the hyperbola.Plugging x = -√15 into the hyperbola equation: ( (√15)² )/9 - y²/6 = 1. That's (15)/9 - y²/6 = 1, which simplifies to 5/3 - y²/6 = 1. Then, subtract 1: 5/3 - 1 = y²/6 => 2/3 = y²/6 => y² = 4 => y = ±2. So, the points of intersection are (-√15, 2) and (-√15, -2). Therefore, the length of this chord is the distance between these two points, which is 4 units.Wait, so is this the minimum length? Let me check. If I take another line through F₁, say with some slope, would the chord length be longer or shorter? I think that the vertical line might give the shortest chord because it's the most "direct" path through the focus, but I'm not entirely sure.Alternatively, maybe the minimal chord is the one that is the latus rectum. Wait, the latus rectum length for a hyperbola is given by 2b²/a. Let me compute that. So, 2*(6)/3 = 4. So, the latus rectum is 4. Wait, that's the same as the vertical chord we just found. So, in this case, the vertical line through F₁ is the latus rectum, and its length is 4.Therefore, the minimal |AB| is 4. So, going back to our earlier equation, |AF₂| + |BF₂| = |AB| + 12. So, the minimal value would be 4 + 12 = 16.Wait, so is 16 the minimal value? Let me just verify this. If I take another chord through F₁, say a horizontal line, but the hyperbola is symmetric about the x-axis, so a horizontal line through F₁ would be y = 0, which is the x-axis itself. But the x-axis intersects the hyperbola at x = ±3, so points would be (3, 0) and (-3, 0). But since we are on the left branch, the intersection points are (-3, 0) and another point? Wait, no, the line y=0 intersects the hyperbola at x = ±3, but since we are on the left branch, only (-3, 0) is on the left. Wait, but the line passes through F₁, which is (-√15, 0). So, if I draw a line from F₁ along the x-axis towards the left, it would only intersect the hyperbola at (-3, 0). Wait, but that's a single point, not two points. So, maybe the line y=0 is tangent to the hyperbola at (-3, 0). Hmm, that might be the case.Wait, let me check. If I plug y=0 into the hyperbola equation, I get x²/9 = 1, so x = ±3. So, the line y=0 intersects the hyperbola at (3, 0) and (-3, 0). So, if I take the line y=0, which passes through F₁ (-√15, 0), it intersects the hyperbola at (-3, 0) and (3, 0). But since we're only considering the left branch, the points A and B would be (-3, 0) and another point? Wait, no, because if the line is y=0, it intersects the left branch only at (-3, 0). So, perhaps in this case, the chord AB is just a single point? That doesn't make sense.Wait, maybe I made a mistake. If the line passes through F₁ and intersects the left branch at two points, then for the line y=0, it intersects the left branch at (-3, 0) and the right branch at (3, 0). But since we are only considering the left branch, maybe we only take (-3, 0) as one point, but then where is the other point? Hmm, perhaps the line y=0 is tangent at (-3, 0), meaning it only intersects at that point. So, in that case, the chord AB would just be a single point, which doesn't make sense for a chord. Therefore, the line y=0 is not a valid secant line for the left branch, only tangent.Therefore, the minimal chord through F₁ on the left branch is indeed the vertical line, which gives us AB = 4. So, |AF₂| + |BF₂| = 4 + 12 = 16.Wait, but let me think again. Is the vertical line the only possible chord? Or is there a shorter chord? Maybe if I take a line with a different slope, the chord AB could be shorter? Hmm, I'm not sure. Let me try to visualize. The hyperbola is symmetric about both axes, so any line through F₁ with a slope would intersect the left branch at two points. The vertical line gives AB = 4, which is the latus rectum. I think in hyperbolas, the latus rectum is indeed the shortest focal chord. So, that should be the minimal AB.Therefore, the minimal value of |AF₂| + |BF₂| is 16.Wait, just to make sure, let me compute |AF₂| + |BF₂| for the points A and B on the vertical line. So, points A and B are (-√15, 2) and (-√15, -2). Let's compute |AF₂| and |BF₂|.F₂ is at (√15, 0). So, the distance from A to F₂ is sqrt[ (√15 - (-√15))² + (0 - 2)² ] = sqrt[ (2√15)² + (-2)² ] = sqrt[ 4*15 + 4 ] = sqrt[60 + 4] = sqrt[64] = 8.Similarly, the distance from B to F₂ is sqrt[ (√15 - (-√15))² + (0 - (-2))² ] = sqrt[ (2√15)² + (2)² ] = sqrt[60 + 4] = sqrt[64] = 8.Therefore, |AF₂| + |BF₂| = 8 + 8 = 16, which matches our earlier conclusion.So, yeah, I think 16 is the correct minimal value.

💡Okay, so I have this problem about a right triangle ABC where angle C is the right angle. The question is asking which of the given equations may not hold true. The options are A, B, C, and D. I need to figure out which one might not be correct.First, let me recall some basic properties of right triangles and trigonometric functions. In a right triangle, the sum of the other two angles is 90 degrees, so angles A and B are complementary. That means angle A plus angle B equals 90 degrees. This is important because it affects the relationships between the sides and the trigonometric functions.Let me label the triangle to make it clearer. Let's say angle C is the right angle, so sides opposite to angles A, B, and C are a, b, and c respectively. So, side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse opposite the right angle C.Now, let's go through each option one by one.**Option A: a = c sin A**I remember that in a right triangle, the sine of an angle is equal to the length of the opposite side divided by the hypotenuse. So, sin A = a/c. If I rearrange this equation, I get a = c sin A. That seems correct because it's just rearranging the definition of sine. So, option A looks true.**Option B: a = b tan A**The tangent of an angle in a right triangle is the opposite side over the adjacent side. So, tan A = a/b. If I rearrange this, I get a = b tan A. That also makes sense because it's another way of expressing the relationship between the sides and the tangent function. So, option B also seems correct.**Option C: c = b / cos B**Let's think about this one. Cosine of an angle is adjacent over hypotenuse. So, cos B = adjacent side / hypotenuse. The adjacent side to angle B is side a, and the hypotenuse is c. So, cos B = a/c. If I rearrange this, I get c = a / cos B. But option C says c = b / cos B. Hmm, that doesn't seem right because the adjacent side to angle B is a, not b. So, this might be incorrect.Wait, let me double-check. If angle B is at vertex B, then the sides adjacent to angle B are side a and side c? No, wait, in a right triangle, each angle has an opposite side and an adjacent side. For angle B, the opposite side is b, and the adjacent side is a. So, cos B = adjacent / hypotenuse = a / c. Therefore, c = a / cos B. But option C says c = b / cos B, which would mean cos B = b / c, but that's actually sin B because sin B = opposite / hypotenuse = b / c. So, cos B is a / c, not b / c. Therefore, option C seems incorrect because it's using side b instead of side a in the expression.**Option D: sin² A + sin² B = 1**Let me verify this identity. I know that in a right triangle, sin A = a/c and sin B = b/c. So, sin² A + sin² B = (a² + b²) / c². But from the Pythagorean theorem, a² + b² = c². Therefore, sin² A + sin² B = c² / c² = 1. So, this identity holds true. Therefore, option D is correct.So, summarizing:- Option A: Correct- Option B: Correct- Option C: Incorrect- Option D: CorrectTherefore, the equation that may not hold true is option C.

💡Okay, so I have this problem about counting the number of unit squares in a specific layout. There's a center square, and then rings around it. The first ring has 8 squares, the second has 16, and so on, adding 8 each time. I need to find the total number of squares from the first ring up to the 50th ring.Alright, let's break this down. First, the center square is just 1 square, right? But the problem doesn't ask about the center; it's only about the rings from the first to the 50th. So I can ignore the center for now.Each ring adds 8 more squares than the previous one. The first ring has 8 squares, the second has 16, which is 8 more than the first, the third would have 24, and so on. So it seems like the number of squares in each ring is increasing by 8 each time.Wait, actually, looking back at the problem statement, it says the first ring has 8 squares, the second has 16, and so on, adding 8 squares for each subsequent ring. So yes, each ring adds 8 more squares than the previous one. That means the number of squares in the nth ring is 8n.So, for the first ring, n=1, it's 8*1=8 squares. For the second ring, n=2, it's 8*2=16 squares, and so on up to the 50th ring, which would be 8*50=400 squares.Now, to find the cumulative number of squares from the first ring up to the 50th ring, I need to sum up all these individual ring squares. That means I need to calculate the sum of 8n for n from 1 to 50.Mathematically, that would be:Sum = 8*1 + 8*2 + 8*3 + ... + 8*50I can factor out the 8:Sum = 8*(1 + 2 + 3 + ... + 50)Now, the sum inside the parentheses is the sum of the first 50 natural numbers. There's a formula for that, right? The sum of the first n natural numbers is given by n(n + 1)/2.So, applying that formula:Sum inside = 50*(50 + 1)/2 = 50*51/2Let me calculate that:50*51 = 2550Then, 2550 divided by 2 is 1275.So, the sum inside the parentheses is 1275.Now, multiplying that by 8:Sum = 8*1275Let me compute that:8*1000 = 80008*200 = 16008*75 = 600Adding those together: 8000 + 1600 = 9600, plus 600 is 10200.So, the cumulative number of unit squares from the first ring up to and including the 50th ring is 10,200.Wait a second, let me double-check my calculations to make sure I didn't make a mistake.First, the formula for the sum of the first n natural numbers is correct: n(n + 1)/2. So for n=50, that's 50*51/2=1275. That seems right.Then, multiplying by 8: 1275*8. Let me do that step by step.1275*8:1000*8=8000200*8=160070*8=5605*8=40Now, adding those together: 8000 + 1600 = 9600, plus 560 is 10160, plus 40 is 10200. Okay, that matches my earlier calculation.So, I think 10,200 is the correct answer.Just to make sure, let's think about the pattern again. Each ring adds 8 more squares than the previous one. So, the number of squares in each ring is an arithmetic sequence with the first term a1=8 and common difference d=8.The formula for the nth term of an arithmetic sequence is a_n = a1 + (n-1)*d. So, for the 50th ring, a50 = 8 + (50-1)*8 = 8 + 49*8 = 8 + 392 = 400. That matches what I thought earlier.The sum of an arithmetic sequence is given by S_n = n/2*(a1 + a_n). So, for n=50, a1=8, a50=400.S_50 = 50/2*(8 + 400) = 25*408 = 10200.Yes, that confirms it again. So, I'm confident that the cumulative number of unit squares from the first ring up to the 50th ring is 10,200.

💡First, determine the age of Darcie's mother. Since Darcie is (4) years old and she is (frac{1}{6}) as old as her mother, the mother's age is:[text{Mother's age} = 4 times 6 = 24 text{ years}]Next, calculate the age of Darcie's grandmother. The mother is (frac{4}{5}) as old as the grandmother, so:[text{Grandmother's age} = frac{24}{frac{4}{5}} = 24 times frac{5}{4} = 30 text{ years}]Then, find the age of Darcie's great-grandfather. The grandmother is (frac{3}{4}) as old as the great-grandfather, thus:[text{Great-grandfather's age} = frac{30}{frac{3}{4}} = 30 times frac{4}{3} = 40 text{ years}]Finally, determine the age of Darcie's great-great-grandmother. The great-grandfather is (frac{7}{10}) as old as the great-great-grandmother, so:[text{Great-great-grandmother's age} = frac{40}{frac{7}{10}} = 40 times frac{10}{7} = frac{400}{7} approx 57.14 text{ years}]

💡To determine the 15th entry in the ordered list of nonnegative integers ( n ) that satisfy ( r_7(3n) leq 3 ), we start by analyzing the condition ( r_7(3n) leq 3 ). This means that ( 3n ) modulo 7 can be 0, 1, 2, or 3. We can express this as:[3n equiv 0, 1, 2, text{ or } 3 pmod{7}]To find the corresponding values of ( n ), we need to solve each congruence:- If ( 3n equiv 0 pmod{7} ), then ( n equiv 0 pmod{7} ).- If ( 3n equiv 1 pmod{7} ), multiplying both sides by the modular inverse of 3 (which is 5, since ( 3 times 5 equiv 1 pmod{7} )) gives ( n equiv 5 pmod{7} ).- If ( 3n equiv 2 pmod{7} ), multiplying both sides by 5 gives ( n equiv 3 pmod{7} ).- If ( 3n equiv 3 pmod{7} ), multiplying both sides by 5 gives ( n equiv 1 pmod{7} ).Thus, the allowable residues of ( n ) modulo 7 are 0, 1, 3, and 5. Listing these residues in order, we get the sequence:[0, 1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, ldots]To find the 15th entry, we observe that the sequence is periodic with a period of 7. Each block of 4 numbers (0, 1, 3, 5) repeats every 7 numbers. Therefore, the 15th entry corresponds to the 15th number in this sequence.Counting through the sequence:1. 02. 13. 34. 55. 76. 87. 108. 129. 1410. 1511. 1712. 1913. 2114. 2215. 24Thus, the 15th entry in the list is 24.

💡First, I need to determine the original number of employees before the workforce reduction. The problem states that after a 27.6% reduction, the company has 462 employees remaining.I know that a 27.6% reduction means that 72.4% of the original workforce remains. This is because 100% - 27.6% = 72.4%.Let’s denote the original number of employees as ( x ). According to the problem, 72.4% of ( x ) equals 462 employees. This can be written as:[0.724 times x = 462]To find ( x ), I need to solve for it by dividing both sides of the equation by 0.724:[x = frac{462}{0.724}]Calculating this gives:[x approx 638.121]Since the number of employees must be a whole number, I round 638.121 to the nearest whole number, which is 638.Therefore, the original number of employees was approximately 638.