Question

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

Answer 1

Let the point be $P(h,k)$

$A(2,3)$ and $B(4,5)$
$PA=\sqrt{{(h-2)}^2+{(k-3)}^2}PB=\sqrt{{(h-4)}^2+{(k+5)}^2}$

Given $PA=PB$

$\sqrt{{(h-2)}^2+{(k-3)}^2}=\sqrt{{(h-4)}^2+{(k-5)}^2}{h}^2+4-4h+k^2+9-6h=h^2+16-8h+k^2+25-10k4h+4k=28h+k=7$

Replacing $h$ by $x$ and $k$ by $y$

$x+y=7$