Complex locus question !?

Slope of AP = (y - y₁) / (x - x₁)

Slope of BP = (y - y₂) / (x - x₂)

Since AP⊥BP, then (Slope of AP) * (Slope of BP) = -1
[(y - y₁) / (x - x₁)] [(y - y₂) / (x - x₂)] = -1
(y - y₁) (y - y₂) = -(x - x₁) (y - y₂)
(y - y₁) (y - y₂) + (x - x₁) (x - x₂) = 0
y² - (y₁ + y₂)y + y₁y₂ + x² - (x₁ + x₂)x + x₁x₂ = 0

The locus of P(x, y) is :
x² + y² - (x₁ + x₂)x - (y₁ + y₂)y + (x₁x₂ + y₁y₂) = 0
This is the equation of a circle.

As angle in semi-circle is 90°, the locus of P can be sketched as a circle with AB as diameter.

Points A and B are fixed, and ∠APB = 90°. Point P must lie on the circle having diameter AB. The locus is every point on the circumference of that circle except A and B themselves.

[(y-y1)/(x-x1)][(y-y2)/(x-x2) = -1
(y-y1)(y-y2)=-(x-x1)(x-x2)
you can continue