A point $P(x, y)$ is such that the sum of squares of its distance from $(a, 0)$ and $(-a, 0)$ is $2 b^2$.…
A point $P(x, y)$ is such that the sum of squares of its distance from $(a, 0)$ and $(-a, 0)$ is $2 b^2$. The equation representing the locus of $P$ is
$x^2+y^2=b^2+a^2$
$x^2+y^2=b^2-a^2$
$x^2+y^2=b^2-2 a^2$
$x^2+y^2=b^2+2 a^2$
Solution
Let the point be $(x, y)$.
$
\begin{aligned}
& \Rightarrow \quad(x-a)^2+(y-0)^2+(x+a)^2+(y-0)^2=2 b^2 \\
& \Rightarrow \quad 2 x^2+2 y^2+2 a^2=2 b^2 \\
& \Rightarrow \quad x^2+y^2=b^2-a^2 .
\end{aligned}
$