answer：Okay, so I have this problem here about sequences. It starts by talking about a geometric sequence {b_n} where the fifth term, b_5, is 2. Then it says that the product of the first nine terms of this sequence is 2^9. Hmm, that makes sense because in a geometric sequence, each term is multiplied by a common ratio. So, if b_5 is 2, then the terms before and after it are related by that ratio.Now, the problem asks about an arithmetic sequence {a_n} with a_5 = 2. It wants the analogous conclusion for this arithmetic sequence. So, I need to figure out what the equivalent statement would be for the arithmetic sequence, similar to how the product of the first nine terms in the geometric sequence is 2^9.First, let me recall what an arithmetic sequence is. In an arithmetic sequence, each term is obtained by adding a common difference to the previous term. So, if I denote the first term as a_1 and the common difference as d, then the nth term is given by a_n = a_1 + (n - 1)d.Given that a_5 = 2, I can write that as:a_5 = a_1 + 4d = 2.Now, for the geometric sequence, the product of the first nine terms is 2^9. Let me think about why that is. In a geometric sequence, the product of terms equidistant from the center is equal to the square of the middle term. Since there are nine terms, the fifth term is the middle one. So, b_1 * b_9 = b_2 * b_8 = ... = b_5^2. Since b_5 is 2, each of these products is 4. There are four such pairs and the middle term itself, so the total product is (4)^4 * 2 = 2^8 * 2 = 2^9. That checks out.For the arithmetic sequence, I need to find an analogous conclusion. Instead of a product, which is multiplicative, in an arithmetic sequence, the analogous operation is additive. So, instead of multiplying terms, we add them. So, maybe the sum of the first nine terms is related to the middle term.In an arithmetic sequence, the sum of terms equidistant from the center is equal to twice the middle term. So, a_1 + a_9 = a_2 + a_8 = ... = 2a_5. Since a_5 is 2, each of these sums is 4. There are four such pairs and the middle term itself, so the total sum is 4 * 4 + 2 = 16 + 2 = 18. Wait, that doesn't seem right because 4 * 4 is 16, plus 2 is 18, but there are nine terms, so maybe I should think differently.Actually, the sum of an arithmetic sequence can be calculated as the number of terms multiplied by the average of the first and last terms. So, the sum S of the first n terms is S = n/2 * (a_1 + a_n). In this case, n = 9, and a_5 is the middle term, which is also the average of the first and ninth terms because in an arithmetic sequence, the middle term is the average of the terms equidistant from it.So, a_5 = (a_1 + a_9)/2. Since a_5 is 2, that means (a_1 + a_9)/2 = 2, so a_1 + a_9 = 4. Similarly, a_2 + a_8 = 4, and so on. There are four such pairs and the middle term a_5. So, the total sum is 4 * 4 + 2 = 18. Wait, but 4 * 4 is 16, plus 2 is 18, but 9 terms should add up to 18? Let me check.Alternatively, using the formula S = n/2 * (a_1 + a_n), with n = 9, and a_1 + a_9 = 4, so S = 9/2 * 4 = 9 * 2 = 18. Yes, that makes sense. So, the sum of the first nine terms is 18.But let me think again. In the geometric sequence, the product was 2^9, which is 512. For the arithmetic sequence, the sum is 18. So, the analogous conclusion is that the sum of the first nine terms is 18.Wait, but in the geometric sequence, the product was 2^9 because each pair multiplied to 4, and there were four pairs and the middle term. For the arithmetic sequence, each pair adds to 4, and there are four pairs and the middle term, so the total sum is 4 * 4 + 2 = 18. That seems consistent.Alternatively, since a_5 is the average of the first nine terms, the sum is 9 * a_5 = 9 * 2 = 18. That's another way to see it. Because in an arithmetic sequence, the average of the terms is equal to the average of the first and last terms, which is also equal to the middle term when the number of terms is odd. So, since a_5 is 2, the average of all nine terms is 2, so the total sum is 9 * 2 = 18.Yes, that makes sense. So, the analogous conclusion is that the sum of the first nine terms is 18.I think that's the answer. Let me just recap to make sure I didn't miss anything. For the geometric sequence, the product of the first nine terms is 2^9 because each pair multiplies to 4, and there are four pairs plus the middle term. For the arithmetic sequence, each pair adds to 4, and there are four pairs plus the middle term, so the total sum is 4 * 4 + 2 = 18. Alternatively, since the average is 2, the sum is 9 * 2 = 18. Both ways, I get 18. So, I'm confident that's the correct answer.

answer：To determine how many integers satisfy the inequality ( |x - 3| leq 7 ), we start by understanding what this inequality means. The absolute value ( |x - 3| ) represents the distance between ( x ) and 3 on the number line. The inequality ( |x - 3| leq 7 ) means that this distance is at most 7 units.To solve the inequality, we can rewrite it as a compound inequality:[-7 leq x - 3 leq 7]Next, we solve for ( x ) by adding 3 to all parts of the inequality:[-7 + 3 leq x - 3 + 3 leq 7 + 3]Simplifying this, we get:[-4 leq x leq 10]This means that ( x ) can be any integer from -4 to 10, inclusive. To find the number of integers in this range, we count the integers from -4 to 10. Starting from -4, the integers are: -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Counting these, we find there are 15 integers in total.Therefore, the number of integers that satisfy the inequality ( |x - 3| leq 7 ) is (boxed{15}).

answer：Alright, so I have this problem about parabolas, midpoints, and angles. Let me try to parse it step by step.First, the problem says that we have a parabola P with vertex V1 and focus F1. Points A and B are on P such that the angle AV1B is 90 degrees. Then, Q is the locus of the midpoint of AB. We're given that P is y = 2x² and V1 is at (0,1). We need to find the ratio F1F2 / V1V2, where Q is another parabola with vertex V2 and focus F2.Wait, hold on. The equation of P is y = 2x², but V1 is at (0,1). Hmm, that seems a bit confusing because usually, the vertex of y = 2x² is at (0,0). Maybe the problem is shifting the vertex? Let me check.If V1 is at (0,1), then the equation of P should be shifted up by 1 unit. So, the standard form of a parabola with vertex at (h,k) is y = a(x - h)² + k. Since h is 0 and k is 1, the equation becomes y = 2x² + 1. But the problem says the equation is y = 2x². Maybe I misread. Let me look again.Wait, the problem says the equation of P is y = 2x², and V1 is at (0,1). That seems contradictory because y = 2x² has its vertex at (0,0). Maybe it's a typo or maybe I need to adjust for that. Perhaps V1 is actually at (0,1), so the equation is y = 2x² + 1? That would make sense because then the vertex is at (0,1). Let me assume that for now because otherwise, the vertex is at (0,0), but the problem says it's at (0,1). So, I think the equation should be y = 2x² + 1.Okay, so P is y = 2x² + 1, vertex V1 at (0,1). Then, points A and B are on P such that angle AV1B is 90 degrees. We need to find the locus of midpoints of AB, which is Q, and then find the ratio of distances between the foci and vertices.Let me denote A as (a, 2a² + 1) and B as (b, 2b² + 1). Then, the midpoint M of AB would be ((a + b)/2, (2a² + 1 + 2b² + 1)/2) = ((a + b)/2, (2a² + 2b² + 2)/2) = ((a + b)/2, a² + b² + 1).Now, since angle AV1B is 90 degrees, the vectors V1A and V1B should be perpendicular. Let's compute these vectors.V1 is (0,1). So, vector V1A is (a - 0, 2a² + 1 - 1) = (a, 2a²). Similarly, vector V1B is (b, 2b²). For these vectors to be perpendicular, their dot product should be zero.So, (a)(b) + (2a²)(2b²) = 0. That is, ab + 4a²b² = 0.Let me factor this: ab(1 + 4ab) = 0.So, either ab = 0 or 1 + 4ab = 0.If ab = 0, then either a = 0 or b = 0. But if a = 0, then point A is (0,1), which is V1, but A is supposed to be on P, so it's allowed, but then angle AV1B would be undefined because A and V1 coincide. Similarly, if b = 0, same issue. So, ab = 0 is not acceptable because it would collapse the angle. So, we must have 1 + 4ab = 0, which gives ab = -1/4.Okay, so ab = -1/4. That's a key relation between a and b.Now, let's go back to the midpoint M. We have M = ((a + b)/2, a² + b² + 1). Let me denote h = (a + b)/2 and k = a² + b² + 1. So, we need to express k in terms of h.We know that (a + b)² = a² + 2ab + b². So, a² + b² = (a + b)² - 2ab.We have ab = -1/4, so a² + b² = (a + b)² - 2*(-1/4) = (a + b)² + 1/2.Therefore, k = (a + b)² + 1/2 + 1 = (a + b)² + 3/2.But h = (a + b)/2, so (a + b) = 2h. Therefore, (a + b)² = 4h².So, substituting back, k = 4h² + 3/2.Thus, the locus Q is the set of points (h, k) such that k = 4h² + 3/2. So, the equation of Q is y = 4x² + 3/2.Wait, but let me check that again. Because earlier, I had k = a² + b² + 1, and a² + b² = (a + b)² - 2ab = (2h)^2 - 2*(-1/4) = 4h² + 1/2. So, k = 4h² + 1/2 + 1 = 4h² + 3/2. Yes, that's correct.So, Q is y = 4x² + 3/2. Therefore, it's a parabola with vertex V2 at (0, 3/2). The standard form is y = 4x² + 3/2, so the vertex is at (0, 3/2).Now, we need to find the focus F2 of Q. For a parabola y = ax² + c, the focus is at (0, c + 1/(4a)). Here, a = 4, so the focus is at (0, 3/2 + 1/(4*4)) = (0, 3/2 + 1/16) = (0, 24/16 + 1/16) = (0, 25/16).Similarly, for the original parabola P, which is y = 2x² + 1, the focus F1 is at (0, 1 + 1/(4*2)) = (0, 1 + 1/8) = (0, 9/8).Wait, hold on. Let me confirm that. The standard form of a parabola y = ax² + c has its focus at (0, c + 1/(4a)). So, for P: y = 2x² + 1, a = 2, c = 1, so focus is at (0, 1 + 1/(4*2)) = (0, 1 + 1/8) = (0, 9/8). Correct.For Q: y = 4x² + 3/2, a = 4, c = 3/2, so focus is at (0, 3/2 + 1/(4*4)) = (0, 3/2 + 1/16) = (0, 24/16 + 1/16) = (0, 25/16). Correct.Now, we need to find the ratio F1F2 / V1V2.First, let's compute F1F2. F1 is at (0, 9/8), F2 is at (0, 25/16). The distance between them is |9/8 - 25/16|.Convert to common denominator: 9/8 = 18/16, so |18/16 - 25/16| = | -7/16 | = 7/16.Next, V1V2. V1 is at (0,1), V2 is at (0, 3/2). The distance is |3/2 - 1| = |1/2| = 1/2.So, the ratio is (7/16) / (1/2) = (7/16) * (2/1) = 7/8.Wait, that's 7/8. But let me double-check my calculations.Wait, F1 is at (0,9/8), which is 1.125, and F2 is at (0,25/16), which is approximately 1.5625. The distance between them is 25/16 - 9/8 = 25/16 - 18/16 = 7/16.V1 is at (0,1), V2 is at (0,3/2) which is 1.5. The distance is 0.5, which is 1/2.So, 7/16 divided by 1/2 is indeed 7/8.Wait, but in the initial problem, the user had a different approach and got 1 as the ratio. Did I make a mistake?Wait, let me see. The user assumed that the equation of P was y = 2x² with vertex at (0,0), but the problem says V1 is at (0,1). So, I think the user made a mistake by not shifting the parabola. I adjusted for that by considering y = 2x² + 1. Maybe that's where the discrepancy is.Wait, let me check the user's solution again.The user wrote:"Assume P is given by y = 2x². Consequently, the vertex V1 = (0,0). Then A = (a, 2a²) and B = (b, 2b²). The midpoint of AB is:[left( frac{a+b}{2}, frac{2a^2 + 2b^2}{2} right) = left( frac{a+b}{2}, a^2 + b^2 right) = left( frac{a+b}{2}, frac{(a+b)^2 + 2ab}{2} right)]Since angle AV1B = 90°, we have ab = -1/2. Therefore, the midpoint becomes:[left( frac{a+b}{2}, frac{(a+b)^2}{2} - 1 right)]Thus, the locus Q of the midpoint is described by y = 2x² - 1.For Q, the vertex V2 is (0, -1). The focus of a parabola y = ax² + bx + c is (0, 1/(4a) + c). Thus, F2 = (0, 1/8 - 1) = (0, -7/8).Now, for V1 = (0, 0) and F1 = (0, 1/(4*2)) = (0, 1/8), the ratio F1F2 / V1V2 becomes:[frac{sqrt{(0 - 0)^2 + (1/8 + 7/8)^2}}{sqrt{(0 - 0)^2 + (0 + 1)^2}} = frac{sqrt{1}}{sqrt{1}} = 1]Thus, the answer is 1."But in the problem statement, it's given that V1 is at (0,1), not (0,0). So, the user incorrectly assumed V1 is at (0,0). Therefore, their entire solution is based on an incorrect assumption. That's why they got a ratio of 1, which might not be correct.In my solution, I correctly accounted for V1 being at (0,1), leading to the equation y = 2x² + 1, and then found Q as y = 4x² + 3/2, resulting in a ratio of 7/8.Wait, but let me double-check my calculations again to be sure.1. Equation of P: y = 2x² + 1, vertex V1 at (0,1).2. Points A(a, 2a² + 1) and B(b, 2b² + 1).3. Midpoint M: ((a + b)/2, (2a² + 2b² + 2)/2) = ((a + b)/2, a² + b² + 1).4. Vectors V1A = (a, 2a²) and V1B = (b, 2b²).5. Dot product: ab + 4a²b² = 0 => ab(1 + 4ab) = 0 => ab = -1/4.6. Midpoint y-coordinate: a² + b² + 1 = (a + b)² - 2ab + 1 = (2h)² - 2*(-1/4) + 1 = 4h² + 1/2 + 1 = 4h² + 3/2.7. So, Q is y = 4x² + 3/2, vertex V2 at (0, 3/2).8. Focus F2: For y = 4x² + 3/2, a = 4, so focus is at (0, 3/2 + 1/(4*4)) = (0, 3/2 + 1/16) = (0, 25/16).9. Focus F1: For y = 2x² + 1, a = 2, so focus is at (0, 1 + 1/(4*2)) = (0, 1 + 1/8) = (0, 9/8).10. Distance F1F2: |25/16 - 9/8| = |25/16 - 18/16| = 7/16.11. Distance V1V2: |3/2 - 1| = 1/2.12. Ratio: (7/16) / (1/2) = 7/8.Yes, that seems correct. So, the ratio is 7/8.Wait, but in the initial problem, the user's solution got 1, but that was because they incorrectly placed V1 at (0,0). So, the correct answer should be 7/8.But let me think again. Maybe I made a mistake in calculating the focus of Q.For a parabola y = ax² + c, the focus is at (0, c + 1/(4a)). So, for Q: y = 4x² + 3/2, a = 4, c = 3/2. So, focus is at (0, 3/2 + 1/(4*4)) = 3/2 + 1/16 = 24/16 + 1/16 = 25/16. Correct.For P: y = 2x² + 1, a = 2, c = 1. Focus is at (0, 1 + 1/(4*2)) = 1 + 1/8 = 9/8. Correct.Distance between F1 and F2: 25/16 - 9/8 = 25/16 - 18/16 = 7/16.Distance between V1 and V2: 3/2 - 1 = 1/2 = 8/16.Wait, hold on. 1/2 is 8/16, and 7/16 divided by 8/16 is 7/8. Yes, that's correct.So, the ratio is 7/8.Wait, but in the initial problem, the user got 1 because they didn't shift the parabola. So, the correct answer is 7/8.But let me check if the equation of Q is correctly derived.We had M = ((a + b)/2, a² + b² + 1). With ab = -1/4.Expressed in terms of h = (a + b)/2, so a + b = 2h.Then, a² + b² = (a + b)^2 - 2ab = 4h² - 2*(-1/4) = 4h² + 1/2.Therefore, k = 4h² + 1/2 + 1 = 4h² + 3/2. So, y = 4x² + 3/2. Correct.Yes, that seems right.So, the ratio is 7/8.But wait, the problem says "determine the ratio F1F2 / V1V2". So, it's the distance between F1 and F2 divided by the distance between V1 and V2.F1 is at (0,9/8), F2 at (0,25/16). Distance is |25/16 - 9/8| = |25/16 - 18/16| = 7/16.V1 is at (0,1), V2 at (0,3/2). Distance is |3/2 - 1| = 1/2.So, 7/16 divided by 1/2 is (7/16) * (2/1) = 7/8.Yes, that's correct.Therefore, the ratio is 7/8.I think that's the correct answer. So, the final answer is 7/8.

answer：Okay, so I have this problem about a moving circle that passes through the point S(2,0) and intercepts a chord of length 4 on the y-axis. There are two parts: first, I need to find the trajectory equation of the moving circle's center C. Second, I have to prove that the sum of λ and μ is a constant value, where λ and μ are defined in terms of vectors involving points A, B, and T on a line l passing through S.Starting with the first part. Let me visualize this. There's a circle that goes through S(2,0), and when it intersects the y-axis, the chord it creates has a length of 4. I need to find the path that the center of this circle takes as it moves.Let me denote the center of the circle as C(x, y). The radius of the circle would then be the distance from C to S, which is sqrt[(x - 2)^2 + (y - 0)^2] = sqrt[(x - 2)^2 + y^2].Now, the circle also intersects the y-axis. The y-axis is the line x = 0. The points where the circle intersects the y-axis can be found by plugging x = 0 into the equation of the circle. The equation of the circle is (x - x)^2 + (y - y)^2 = r^2, but wait, that's not right. Let me correct that. The general equation of a circle with center (h, k) is (x - h)^2 + (y - k)^2 = r^2. So, substituting x = 0, we get (0 - x)^2 + (y - y)^2 = r^2, which simplifies to x^2 + (y - y)^2 = r^2. Hmm, that seems off. Wait, no, if the center is (x, y), then the equation is (X - x)^2 + (Y - y)^2 = r^2. So, plugging in X=0, we get (0 - x)^2 + (Y - y)^2 = r^2, which is x^2 + (Y - y)^2 = r^2.So, solving for Y, we have (Y - y)^2 = r^2 - x^2. Therefore, Y = y ± sqrt(r^2 - x^2). The two points where the circle intersects the y-axis are (0, y + sqrt(r^2 - x^2)) and (0, y - sqrt(r^2 - x^2)). The distance between these two points is the length of the chord, which is given as 4.So, the distance between (0, y + sqrt(r^2 - x^2)) and (0, y - sqrt(r^2 - x^2)) is 2*sqrt(r^2 - x^2). This equals 4. So, 2*sqrt(r^2 - x^2) = 4. Dividing both sides by 2, sqrt(r^2 - x^2) = 2. Squaring both sides, r^2 - x^2 = 4. Therefore, r^2 = x^2 + 4.But we know that r is also the distance from the center (x, y) to the point S(2,0). So, r^2 = (x - 2)^2 + y^2. Therefore, we have (x - 2)^2 + y^2 = x^2 + 4.Let me expand (x - 2)^2: that's x^2 - 4x + 4. So, substituting back, we get x^2 - 4x + 4 + y^2 = x^2 + 4. Subtracting x^2 and 4 from both sides, we have -4x + y^2 = 0. So, y^2 = 4x.Therefore, the trajectory of the center C is the parabola y^2 = 4x.Okay, that seems straightforward. Let me double-check. If the center is (x, y), the radius squared is (x - 2)^2 + y^2. The chord on the y-axis has length 4, so the distance between the two intersection points is 4. That gives us 2*sqrt(r^2 - x^2) = 4, leading to sqrt(r^2 - x^2) = 2, so r^2 = x^2 + 4. Equating that to (x - 2)^2 + y^2, expanding, and simplifying gives y^2 = 4x. Yep, that looks correct.Now, moving on to the second part. I need to prove that the sum of λ and μ is a constant, where λ and μ are defined in terms of vectors. Let me parse the definitions again.Line l passes through point S and intersects the curve C (which is the parabola y^2 = 4x) at points A and B. It also intersects the y-axis at point T. The vectors TA and TB are defined as λ times SA and μ times SB, respectively. So, vectorially, TA = λ SA and TB = μ SB.I need to show that λ + μ is a constant, regardless of the specific line l chosen.First, let me try to understand the vector definitions. Vector TA is from T to A, and vector SA is from S to A. Similarly, vector TB is from T to B, and vector SB is from S to B.So, TA = λ SA implies that vector from T to A is λ times the vector from S to A. Similarly for TB and SB.This suggests that points T, S, A, and B are related in some way. Maybe T is a point such that it's scaled along the line from S to A and S to B.Let me try to model this mathematically.Let me denote the coordinates:- S is (2, 0).- Let line l pass through S(2,0) and intersect the parabola C at points A and B.- Line l also intersects the y-axis at point T, which is (0, t) for some t.I need to express vectors TA and SA, and TB and SB.Let me denote point A as (x1, y1) and point B as (x2, y2). Since A and B lie on the parabola y^2 = 4x, we have y1^2 = 4x1 and y2^2 = 4x2.Line l passes through S(2,0) and T(0, t). So, the parametric equation of line l can be written as:x = 2 - 2sy = 0 + t swhere s is a parameter. Alternatively, in slope-intercept form, the line passes through (2,0) and (0, t), so the slope m is (t - 0)/(0 - 2) = -t/2.So, the equation of line l is y = (-t/2)(x - 2). Simplifying, y = (-t/2)x + t.Alternatively, in parametric form, we can write:x = 2 - 2ky = 0 + t kwhere k is a parameter.But maybe it's easier to use the slope-intercept form. So, y = (-t/2)x + t.We can find the points A and B where this line intersects the parabola y^2 = 4x.Substituting y from the line into the parabola equation:[(-t/2)x + t]^2 = 4xLet me expand the left side:[(-t/2 x + t)]^2 = ( (-t/2 x)^2 + 2*(-t/2 x)(t) + t^2 ) = (t^2/4 x^2 - t^2 x + t^2)So, t^2/4 x^2 - t^2 x + t^2 = 4xBring all terms to one side:t^2/4 x^2 - t^2 x + t^2 - 4x = 0Multiply both sides by 4 to eliminate the fraction:t^2 x^2 - 4t^2 x + 4t^2 - 16x = 0Let me rearrange terms:t^2 x^2 + (-4t^2 - 16) x + 4t^2 = 0This is a quadratic in x:t^2 x^2 - (4t^2 + 16) x + 4t^2 = 0Let me factor out t^2:t^2 (x^2 - 4x) - 16x + 4t^2 = 0Wait, maybe it's better to just keep it as is and use the quadratic formula.Let me denote the quadratic as:A x^2 + B x + C = 0Where:A = t^2B = -4t^2 - 16C = 4t^2So, the solutions are:x = [4t^2 + 16 ± sqrt( (4t^2 + 16)^2 - 16 t^4 ) ] / (2 t^2)Let me compute the discriminant D:D = (4t^2 + 16)^2 - 16 t^4Expand (4t^2 + 16)^2:= 16t^4 + 128t^2 + 256So, D = 16t^4 + 128t^2 + 256 - 16t^4 = 128t^2 + 256Factor out 128:D = 128(t^2 + 2)So, sqrt(D) = sqrt(128(t^2 + 2)) = 8 sqrt(2(t^2 + 2))Wait, sqrt(128) is 8*sqrt(2), so sqrt(128(t^2 + 2)) = 8 sqrt(2(t^2 + 2)).Therefore, the solutions for x are:x = [4t^2 + 16 ± 8 sqrt(2(t^2 + 2)) ] / (2 t^2)Simplify numerator:Factor out 4:= [4(t^2 + 4) ± 8 sqrt(2(t^2 + 2)) ] / (2 t^2)Divide numerator and denominator by 2:= [2(t^2 + 4) ± 4 sqrt(2(t^2 + 2)) ] / t^2So, x = [2(t^2 + 4) ± 4 sqrt(2(t^2 + 2)) ] / t^2This seems complicated. Maybe there's a better way to approach this.Alternatively, perhaps using parametric equations for the parabola. For the parabola y^2 = 4x, a parametric representation is x = at^2, y = 2at. Here, comparing to y^2 = 4x, we have a = 1, so x = t^2, y = 2t.So, any point on the parabola can be written as (t^2, 2t). Let me use this parametrization.Let me denote points A and B as (t1^2, 2t1) and (t2^2, 2t2), respectively.Since both A and B lie on line l, which passes through S(2,0) and T(0, t). So, the line l can be parametrized as passing through S(2,0) and having direction vector towards T(0,t). So, the parametric equations can be written as:x = 2 - 2sy = 0 + t swhere s is a parameter.Alternatively, the slope of line l is (t - 0)/(0 - 2) = -t/2, so the equation is y = (-t/2)(x - 2).So, y = (-t/2)x + t.Now, since point A(t1^2, 2t1) lies on this line, substituting into the equation:2t1 = (-t/2)(t1^2) + tSimilarly, for point B(t2^2, 2t2):2t2 = (-t/2)(t2^2) + tSo, we have:2t1 = (-t/2) t1^2 + tand2t2 = (-t/2) t2^2 + tLet me rearrange these equations:For point A:(-t/2) t1^2 + t - 2t1 = 0Multiply both sides by 2 to eliminate the fraction:- t t1^2 + 2t - 4t1 = 0Similarly, for point B:- t t2^2 + 2t - 4t2 = 0So, both t1 and t2 satisfy the quadratic equation:- t t^2 + 2t - 4t = 0Wait, that seems off. Let me correct that.Wait, for point A:(-t/2) t1^2 + t - 2t1 = 0Multiply by 2:- t t1^2 + 2t - 4t1 = 0Similarly for point B:- t t2^2 + 2t - 4t2 = 0So, both t1 and t2 satisfy the equation:- t t^2 + 2t - 4t = 0Wait, that can't be right because the equation is in terms of t1 and t2, not t. Let me clarify.Let me denote the parameter as s, so the equation is:- t s^2 + 2t - 4s = 0So, for both t1 and t2, we have:- t t1^2 + 2t - 4t1 = 0and- t t2^2 + 2t - 4t2 = 0So, t1 and t2 are roots of the quadratic equation:- t s^2 + 2t - 4s = 0Let me write it as:t s^2 + 4s - 2t = 0Wait, no, original equation after multiplying by 2:- t s^2 + 2t - 4s = 0So, rearranged:t s^2 + 4s - 2t = 0Wait, no, let me check:Original equation after multiplying by 2:- t s^2 + 2t - 4s = 0So, bringing all terms to one side:t s^2 - 2t + 4s = 0Factor t:t(s^2 - 2) + 4s = 0Hmm, not sure if that helps. Alternatively, write it as:t s^2 + 4s - 2t = 0Yes, that's correct.So, the quadratic equation in s is:t s^2 + 4s - 2t = 0Therefore, the roots s = t1 and s = t2 satisfy:t1 + t2 = -4/tandt1 t2 = -2t / t = -2Wait, using quadratic formula, for equation a s^2 + b s + c = 0, sum of roots is -b/a, product is c/a.Here, a = t, b = 4, c = -2t.So, sum of roots t1 + t2 = -4/tProduct t1 t2 = (-2t)/t = -2Okay, so t1 + t2 = -4/t and t1 t2 = -2.Now, I need to relate this to vectors TA and SA, and TB and SB.Given that TA = λ SA and TB = μ SB.Let me express these vectors.First, point T is (0, t).Point A is (t1^2, 2t1)Point S is (2, 0)So, vector TA is A - T = (t1^2 - 0, 2t1 - t) = (t1^2, 2t1 - t)Vector SA is A - S = (t1^2 - 2, 2t1 - 0) = (t1^2 - 2, 2t1)Given that TA = λ SA, so:(t1^2, 2t1 - t) = λ (t1^2 - 2, 2t1)Therefore, component-wise:t1^2 = λ (t1^2 - 2) ...(1)2t1 - t = λ (2t1) ...(2)Similarly, for point B:Vector TB = (t2^2, 2t2 - t)Vector SB = (t2^2 - 2, 2t2)Given TB = μ SB:(t2^2, 2t2 - t) = μ (t2^2 - 2, 2t2)So:t2^2 = μ (t2^2 - 2) ...(3)2t2 - t = μ (2t2) ...(4)Let me solve equations (1) and (2) for λ.From equation (1):t1^2 = λ (t1^2 - 2)So, λ = t1^2 / (t1^2 - 2)From equation (2):2t1 - t = λ (2t1)Substitute λ from above:2t1 - t = (t1^2 / (t1^2 - 2)) * 2t1Multiply both sides by (t1^2 - 2):(2t1 - t)(t1^2 - 2) = 2t1 * t1^2Expand left side:2t1(t1^2 - 2) - t(t1^2 - 2) = 2t1^3 - 4t1 - t t1^2 + 2tRight side: 2t1^3So, set equal:2t1^3 - 4t1 - t t1^2 + 2t = 2t1^3Subtract 2t1^3 from both sides:-4t1 - t t1^2 + 2t = 0Rearrange:- t t1^2 -4t1 + 2t = 0Multiply both sides by -1:t t1^2 + 4t1 - 2t = 0Which is the same quadratic equation as before: t t1^2 + 4t1 - 2t = 0So, this is consistent.Similarly, for μ, we can derive:From equation (3):t2^2 = μ (t2^2 - 2)So, μ = t2^2 / (t2^2 - 2)From equation (4):2t2 - t = μ (2t2)Substitute μ:2t2 - t = (t2^2 / (t2^2 - 2)) * 2t2Multiply both sides by (t2^2 - 2):(2t2 - t)(t2^2 - 2) = 2t2 * t2^2Expand left side:2t2(t2^2 - 2) - t(t2^2 - 2) = 2t2^3 - 4t2 - t t2^2 + 2tRight side: 2t2^3Set equal:2t2^3 - 4t2 - t t2^2 + 2t = 2t2^3Subtract 2t2^3:-4t2 - t t2^2 + 2t = 0Which is the same quadratic equation: t t2^2 + 4t2 - 2t = 0So, consistent again.Now, I need to find λ + μ.From above, λ = t1^2 / (t1^2 - 2)Similarly, μ = t2^2 / (t2^2 - 2)So, λ + μ = [t1^2 / (t1^2 - 2)] + [t2^2 / (t2^2 - 2)]Let me compute this sum.First, let me note that t1 and t2 are roots of the quadratic equation t s^2 + 4s - 2t = 0, so:t1 + t2 = -4/tt1 t2 = -2I need to express λ + μ in terms of t1 and t2, and then use these relations to find a constant.Let me compute λ + μ:λ + μ = [t1^2 / (t1^2 - 2)] + [t2^2 / (t2^2 - 2)]Let me combine these fractions:= [t1^2 (t2^2 - 2) + t2^2 (t1^2 - 2)] / [(t1^2 - 2)(t2^2 - 2)]Simplify numerator:= t1^2 t2^2 - 2t1^2 + t1^2 t2^2 - 2t2^2= 2 t1^2 t2^2 - 2(t1^2 + t2^2)Denominator:= (t1^2 - 2)(t2^2 - 2) = t1^2 t2^2 - 2t1^2 - 2t2^2 + 4So, numerator: 2 t1^2 t2^2 - 2(t1^2 + t2^2)Denominator: t1^2 t2^2 - 2(t1^2 + t2^2) + 4Let me denote N = numerator, D = denominator.So, N = 2 t1^2 t2^2 - 2(t1^2 + t2^2)D = t1^2 t2^2 - 2(t1^2 + t2^2) + 4Notice that N = 2(t1^2 t2^2 - (t1^2 + t2^2))And D = t1^2 t2^2 - 2(t1^2 + t2^2) + 4Let me express N and D in terms of t1 + t2 and t1 t2.We know that:t1 + t2 = -4/tt1 t2 = -2Let me compute t1^2 + t2^2:t1^2 + t2^2 = (t1 + t2)^2 - 2 t1 t2 = (16/t^2) - 2*(-2) = 16/t^2 + 4Similarly, t1^2 t2^2 = (t1 t2)^2 = (-2)^2 = 4So, substituting into N:N = 2(4 - (16/t^2 + 4)) = 2(4 - 16/t^2 - 4) = 2(-16/t^2) = -32/t^2Similarly, D:D = 4 - 2(16/t^2 + 4) + 4 = 4 - 32/t^2 - 8 + 4 = (4 - 8 + 4) - 32/t^2 = 0 - 32/t^2 = -32/t^2So, N = -32/t^2 and D = -32/t^2Therefore, λ + μ = N/D = (-32/t^2)/(-32/t^2) = 1So, λ + μ = 1, which is a constant.Therefore, the sum of λ and μ is always 1, regardless of the line l chosen.That's interesting. So, no matter how we choose the line l through S intersecting the parabola, the sum of these scaling factors λ and μ remains constant at 1.Let me recap to ensure I didn't make any mistakes.1. Found the equation of the circle's center by considering the chord length on the y-axis, leading to y^2 = 4x.2. For the second part, parametrized the parabola, expressed the line l in terms of t, found the points A and B in terms of parameters t1 and t2, derived the quadratic equation they satisfy, then expressed λ and μ in terms of t1 and t2, and through algebraic manipulation, showed that λ + μ = 1.Everything seems consistent. I don't see any errors in the steps. The key was recognizing the parametric form of the parabola and using the properties of the roots of the quadratic equation to express λ and μ in terms that could be combined and simplified to a constant.

answer：Alright, so I have two math problems to solve here. The first one is a calculation involving exponents, negative numbers, and absolute values. The second one is a system of equations that I need to solve. Let me tackle them one by one.Starting with the first problem: Calculate (left(-3right)^{2} times 3^{-1} + left(-5 + 2right) + |-2|).Okay, let's break this down step by step. First, I need to compute each part separately and then combine them. I remember that exponents come before multiplication and division in the order of operations, so I'll start with (left(-3right)^{2}).Calculating (left(-3right)^{2}): When you square a negative number, the result is positive. So, (left(-3right)^{2} = 9).Next, I have (3^{-1}). Negative exponents mean reciprocals, so (3^{-1} = frac{1}{3}).Now, multiplying these two results: (9 times frac{1}{3}). That should be straightforward. Multiplying 9 by 1/3 is the same as dividing 9 by 3, which equals 3.Moving on to the next part: (left(-5 + 2right)). Adding a negative number is like subtracting. So, (-5 + 2) is the same as (2 - 5), which equals (-3).Then, we have the absolute value part: (|-2|). Absolute value makes any number inside it positive, so (|-2| = 2).Now, putting it all together: (3 + (-3) + 2). Let's compute this step by step. First, (3 + (-3)) is (0), and then (0 + 2) equals (2).Wait, that seems too simple. Did I do that right? Let me double-check each step. 1. (left(-3right)^{2} = 9) – correct.2. (3^{-1} = frac{1}{3}) – correct.3. (9 times frac{1}{3} = 3) – correct.4. (-5 + 2 = -3) – correct.5. (|-2| = 2) – correct.6. Adding them up: (3 + (-3) + 2 = 2) – yes, that's correct.Okay, so the first problem seems to be done. The result is 2.Now, onto the second problem: Solve the system of equations:[begin{cases}2x - y = 3 quad text{①} x + y = 6 quad text{②}end{cases}]I need to find the values of (x) and (y) that satisfy both equations simultaneously. There are a couple of methods to solve systems of equations: substitution and elimination. I think elimination might be straightforward here because if I add the two equations, the (y) terms will cancel out.Let's try adding equation ① and equation ②:[(2x - y) + (x + y) = 3 + 6]Simplifying the left side:(2x - y + x + y = 3x) (since (-y + y = 0))And the right side:(3 + 6 = 9)So, we have:(3x = 9)To solve for (x), divide both sides by 3:(x = 3)Now that we have (x = 3), we can substitute this value back into one of the original equations to find (y). Let's use equation ② because it looks simpler:(x + y = 6)Substituting (x = 3):(3 + y = 6)Subtract 3 from both sides:(y = 3)So, both (x) and (y) are 3. Let me verify this solution in both original equations to make sure it works.Checking equation ①:(2x - y = 3)Substituting (x = 3) and (y = 3):(2(3) - 3 = 6 - 3 = 3)Which matches the right side.Checking equation ②:(x + y = 6)Substituting (x = 3) and (y = 3):(3 + 3 = 6)Which also matches the right side.Great, so the solution (x = 3) and (y = 3) satisfies both equations. That means I've solved the system correctly.In summary, for the first problem, the calculation simplifies to 2, and for the second problem, the solution to the system is (x = 3) and (y = 3).

answer：First, I need to determine the population of the town after the first year. The population increased by 22%, so I calculate 22% of 800 and add it to the original population.Next, I know the population after the second year is 1220. This means that the population increased by an unknown percentage from the population at the end of the first year. I set up an equation to represent this growth.To find the unknown percentage increase, I divide both sides of the equation by the population after the first year to isolate the growth factor. Then, I subtract 1 to solve for the percentage increase.Finally, I convert the decimal result back to a percentage to find the percentage increase in the second year.