Question

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

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

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.
Here, $A\equiv (-a,0)$ and $B\equiv (a,0)$. 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.