Find the equation to the locus of a point which is always equidistant from the points whose coordinates are $(2, 3)$ and $(4, 5)$

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Solution

Let the point be $P (h, k)$
$A (2, 3)$ and $B (4, 5)$
$P A = \sqrt{{(h - 2)}^{2} + {(k - 3)}^{2}} P B = \sqrt{{(h - 4)}^{2} + {(k + 5)}^{2}}$
Given $P A = P B$
$\sqrt{{(h - 2)}^{2} + {(k - 3)}^{2}} = \sqrt{{(h - 4)}^{2} + {(k - 5)}^{2}} h^{2} + 4 - 4 h + k^{2} + 9 - 6 h = h^{2} + 16 - 8 h + k^{2} + 25 - 10 k 4 h + 4 k = 28 h + k = 7$
Replacing $h$ by $x$ and $k$ by $y$
$x + y = 7$

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