A point P moves so that AP and BP are perpendicular. Find the equation of the locus of P

Question

A point p(x,y) moves so that AP and BP are perpendicular, given A=(3,2) and B =(-4,1). Find the equation of the locus of P.

Can someone please advise me on what to do for this question. Just need a direction. Thanks.

Answer 1

Direction: The locus is the circle with diameter $AB$.

Answer 2

At each stage, and since the product of slopes of perpendicular lines is $\;-1\;$ , we get:

$$\frac{y-2}{x-3}=-\frac{x+4}{y-1}\iff y^2-3y+2=-x^2-x+12\iff$$

$$\iff\left(x+\frac12\right)^2+\left(y-\frac32\right)^2-\frac52=10\;\;\ldots\ldots$$