Question

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

• A. $(x^2 + y^2)^2 = 4Rx^2y^2$
• B. $(x^2 + y^2)(x+y) = R^2xy$
• C. $(x^2 + y^2)^3 = 4R^2x^2y^2$
• D. $(x^2 + y^2)^2 = 4R^2x^2y^2$

Answer 1

Answer: (C)

$(x^2 + y^2)^3 = 4R^2x^2y^2$

Answer 2

Slope of $AB = \frac{-h}{k}$
Equation of $AB$ is $hx + ky = h^2 + k^2$
$A\bigg(\frac{h^2 + k^2}{h}, 0 \bigg), B\bigg(0, \frac{h^2 + k^2}{k}\bigg)$
$AB = 2R$
$\Rightarrow \, (h^2 + k^2)^3 = 4R^2h^2k^2$
$\Rightarrow \, (x^2 + y^2)^3 \, = 4R^2x^2y^2$