Question

A (−a,0) and B (a,0) are two fixed points. The locus of a point P such as $\angle APB$ is a right angle, is

• A. $x^2$+ $y^2$ = $a^2$
• B. $x^2$- $y^2$ = $a^2$
• C. ${2x}^2$+ $y^2$ = $a^2$
• D. pair of lines

Answer 1

The correct option is A $x^2$+ $y^2$ = $a^2$

Let P (x, y) be the moving point whose locus is required. $LetA\equiv (-a,0)andB\equiv (a,0),Given\angle APB=\frac{\pi }{2}\therefore A{P}^2+P{B}^2=A{B}^2$

$\Rightarrow (x+a{)}^2+y^2+(x-a{)}^2+y^2=(2a{)}^2\Rightarrow 2x^2+2y^2+2a^2=4a^2\Rightarrow x^2+y^2=a^2$
Which is the required locus.
Hence, (a) is the correct answer.