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. a hyperbola
• C. an ellipse
• D. a parabola

Answer 1

Answer: (A)

a straight line

Answer 2

Let A = (1, 4) and B = (1, -5). The midpoint of AB is M = $(1, -1/2)$. By Apollonius' theorem, $PA^2 + PB^2 = 2(PM^2 + AM^2)$. To maximize $PA^2 + PB^2$, we need to maximize $PM$. The circle $(x - 1)^2 + (y - 1)^2 = 1$ has center C(1, 1) and radius $r=1$. The distance $PM$ is maximized when P is on the line CM and farthest from M. The line CM is the vertical line $x=1$. The points on the circle with $x=1$ are found by $(1-1)^2 + (y-1)^2 = 1 \implies (y-1)^2 = 1 \implies y=0$ or $y=2$. The points are P1(1, 0) and P2(1, 2). M is (1, -1/2). P2(1, 2) is farther from M than P1(1, 0). So P is (1, 2). The points A(1, 4), B(1, -5), and P(1, 2) all lie on the line $x=1$.