First slide

Cartesian plane

Question

The coordinates of two points A and Bare ( 3, 4) and (5, - 2) respectively. If P is a point not lying on any of the coordinate axes such that $P A = P B$ and Area of of $∆ P A B = 10$ , then the coordinates of P, are

Moderate

A

(2, 7)
B

(7, 2)
C

(1,0)
D

(0, 1)

Solution

Let the coordinates of P be (x, y). Then,
$\begin{array}{l} P A = P B \\ \Rightarrow P A^{2} = P B^{2} \\ \Rightarrow (x - 3)^{2} + (y - 4)^{2} = (x - 5)^{2} + (y + 2)^{2} \\ x - 3 y - 1 = 0 . . . (i) \end{array}$
Now,
Area of $△ P A B = 10$
$\begin{array}{l} \Rightarrow \frac{1}{2} |\begin{array}{rrr} x & y & 1 \\ 3 & 4 & 1 \\ 5 & - 2 & 1 \end{array}| = \pm 10 \\ \Rightarrow 6 x + 2 y - 26 = \pm 20 \end{array}$
Solving (i) with equations in (ii), we obtain $x = 7, y = 2$ and $x = 1, y = 0$ .
But, P does not lie on any of the coordinate axes. Thus, the coordinates of P are $(7, 2)$

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