Question

Find the equation of the locus of all points equidistant from the point $(2,4)$ and the $y$-axis.

Answer 1

Let (x, y) be the unknown point. Its distance from y-axis is |x|. It's distance from $(2,4)$ is given by

$d=\left(\right(x-2{)}^2+(y-4{)}^2{)}^{1/2}$

Now put $d=|x|$, since point is equidistant from the given point and y-axis.

$|x|=\left(\right(x-2{)}^2+(y-4{)}^2{)}^{1/2}$

$x^2=(x-2{)}^2+(y-4{)}^2$

$x^2=x^2-4x+4+(y-4{)}^2$

$(y-4{)}^2-4x+4=0$

So the locus of the given point is a parabola.