A line cuts the x - axis at A(7, 0) and the y - axis at B(0, -5). A variable line PQ is drawn perpendicular to AB. Cutting the x - axis at P and the y - axis at Q.
If AQ and BP intersect at R, the locus of R is
x2+y2−7x+5y=0
Let P(a, 0) and Q(0, b)
Slope of PQ =−ba
−ba×57=−1⇒ ab=57
Equation of AQ is x7+yb=1 ⇒ b=7y7−x
Equation of BP is xa−y5=1 ⇒ a=5x5+y
so that 5x5+y×7−x7y=ab=57⇒ x(7−x)=y(5+y)
⇒ x2+y2−7x+5y=0
which is the locus of R(x, y).