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A tangent to the hyperbola x^2 25 - y^2 9 = 1 intersects the coordinate axes at points A and B . Let C be the locus of the midpoint of the line segment AB . If the curve C intersects the line y=x at a point P in the first quadrant, then the square of the distance of P from the origin is ______.

Correct answer

8

Step-by-step solution

The equation of the tangent to the hyperbola in parametric form is: x 5 - y 3 = 1 This tangent intersects the x-axis at A(5 , 0) and the y-axis at B(0, -3 ) . Let (h, k) be the midpoint of AB . Then: h = 5 2 = 2h 5 = 5 2h k = - 3 2 = - 2k 3 = - 3 2k Using the identity ^2 - ^2 = 1 , we have: ( 5 2h )^2 - (- 3 2k )^2 = 1 25 4h^2 - 9 4k^2 = 1 So, the locus C is 25 4x^2 - 9 4y^2 = 1 . To find the intersection of C with the line y=x , substitute y=x into the locus equation: 25 4x^2 - 9 4x^2 = 1 16 4x^2 = 1 4x^2 = 16 x^2

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