Analytic geometry, distances

Question

Find the equation of the geometric place:

Whose distance to the point $(4,0)$ equals half the distance to the straight line $x=19$

Im using the formula for distance between points $P(4,0), Q(19,0)$ and an arbitrary line $l: d(l,P)=d(l,Q)/2$ but it gets me nowhere. Am I doing it right?

Answer 1

Hints:

• $d((x,y),(4,0)) = \sqrt{(x-4)^2+y^2}$;

• If $\ell: x = 19$, then $d((x,y),\ell) = |x-19| = \sqrt{(x-19)^2}$;

• You want to look at $$d((x,y),(4,0)) = \frac{1}{2}d((x,y),\ell).$$