The sum of the squares of the distances of a moving point from two fixed points $(a, 0)$ and $(- a, 0)$ is equal to a constant quantity $2 c^{2}$ . The equation of its locus is given by $h^{2} + k^{2} = m c^{2} - a^{2}$ , where $m$ is a positive constant, then find the value of $m$ .

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Solution

Let $P (h, k)$ be any position of the moving point and let $A (a, 0), B (- a, 0)$ be the given points.
Given, ${P A}^{2} + {P B}^{2} = 2 c^{2}$
${(h a)}^{2} + {(k 0)}^{2} + {(h + a)}^{2} + {(k 0)}^{2} = 2 c^{2}$
$h^{2} 2 a h + a^{2} + k^{2} + h^{2} + 2 a h + a^{2} + k^{2} = 2 c^{2}$
$2 h^{2} + 2 k^{2} + 2 a^{2} = 2 c^{2}$
$h^{2} + k^{2} = c^{2} - a^{2}$
Hence, the locus of $(h, k)$ is $h^{2} + k^{2} = c^{2} - a^{2}$

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