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Question

The point P = (-1,0) lies on the ellipse 4x^2 + y^2 = 4. Let Q be another point on this ellipse, and let d be the maximum length of PQ. Find d^2.

Accepted Answer

Let Q = (x,y). From the given information, y^2 = 4 - 4x^2. Therefore, align* PQ^2 &= (x + 1)^2 + y^2 &= x^2 + 2x + 1 + 4 - 4x^2 &= -3x^2 + 2x + 5 &= -3 ( x - 13 )^2 + 163. align*This is maximized when x = 13, and d^2 = 163.