Question

Let \(A(1, 4)\) and \(B(1, -5)\) be two points. Let \(P\) be a point on the circle \((x - 1)^2 + (y - 1)^2 = 1\) such that \((PA)^2 + (PB)^2\) have maximum value, then the points, \(P\), \(A\) and \(B\) lie on :

• A. a straight line
• B. an ellipse
• C. a parabola
• D. a hyperbola

Hint:

Use \(PA^2 + PB^2 = 2PM^2 + \frac{AB^2}{2}\) with \(M\) the midpoint of \(AB\); it is maximum when \(P\) is farthest from \(M\) on the circle.

Answer 1

The Correct Option is A

Midpoint of \(AB\) is \(M = (1, -\tfrac12)\); \(AB = 9\) is fixed.
By the median (Apollonius) formula: \(PA^2 + PB^2 = 2PM^2 + \frac{AB^2}{2}\).
So we need \(PM\) maximum for \(P\) on the circle with centre \(C(1,1)\), radius 1.
\(M\), \(C\) lie on the line \(x = 1\); the farthest point of the circle from \(M\) is on this line beyond \(C\): \(P = (1, 2)\).
\(A(1,4)\), \(B(1,-5)\), \(P(1,2)\) all have \(x = 1\), so they lie on a straight line.
Answer: (1) a straight line