Equation of locus Point P$(x, y)$ moves in such a way that its distance from the point $(3, 5)$ is proportional to its distance from the point $(-2, 4)$. Find the locus of P if the origin is a point on the locus.
Answer:
$$(x-3)^2 + (y-5)^2 =  (x+2)^2 + (y-4)^2$$
or, $$10x+2y-14=0$$
or, $$5x+y-7=0$$
but answer given is $$7x^2+7y^2+128x-36y=0$$

$$(x-3)^2 + (y-5)^2 =  \lambda((x+2)^2 + (y-4)^2).$$
We express that the curve passes through the orgin:
$$(-3)^2 + (-5)^2 =  \lambda((+2)^2 + (-4)^2),$$
hence
$$\lambda=\frac{17}{10}.$$
