Find the locus of sum of the distance of a moving point from the points $(4, 0)$ and $(- 4, 0)$ is always $10$ units.

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Solution

Let the point be $(h, k)$
then ,
$\sqrt{(h - 4)^{2} + k^{2}} + \sqrt{(h + 4)^{2} + k^{2}} = 10$
$=> \sqrt{(h - 4)^{2} + k^{2}} = 10 - \sqrt{(h + 4)^{2} + k^{2}}$
Squaring both side,
$=> h^{2} + 16 - 8 h + k^{2} = 100 + h^{2} + 16 + 8 h + k^{2} - 20 \sqrt{(h + 4)^{2} + k^{2}}$
solving and squaring again,
$=> 25 (h^{2} + 16 + 8 h + k^{2}) = 16 h^{2} + 200 h + 625$
$=> 9 h^{2} + 25 k^{2} = 225$
therefore,Locus is a Ellipse.
$\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$

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