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

Engineering

Mathematics

A Circle

Question

(x² + y²)³ = 4 R²x²y²

(x² + y²)(x + y) = R²xy

(x² + y²)² = 4 R²x²y²

(x² + y²)2 = 4 Rx²y²

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$Slope of AB = \frac{- h}{k}$

Equation of AB is hx + hy = h² + k²

$A (\frac{h^{2} + k^{2}}{h}, 0), B (0, \frac{h^{2} + k^{2}}{k})$

As, AB = 2R

⇒ (h² + k²)³ = 4R²h²k²

⇒ (x² + y²)³ = 4R²x²y²