Question

Find the equation to the locus of a point which is always equidistant from the points whose coordinates are $(1,0)$ and $(0,-2)$.

Answer 1

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

$A(1,0)$ and $B(0,-2)$

$PA=\sqrt{{(h-1)}^2+{(k-0)}^2}PB=\sqrt{{(h-0)}^2+{(k-(-2\left)\right)}^2}=\sqrt{{(h-0)}^2+{(k+2)}^2}$

Given $PA=PB$

$\sqrt{{(h-1)}^2+{(k-0)}^2}=\sqrt{{(h-0)}^2+{(k+2)}^2}{h}^2+1-2h+k^2=h^2+k^2+4+4k4k+2h+3=0$

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

$4y+2x+3=0$

is the required locus.