Find the locus of the feet of the normals drawn to the family of circles $x^{2} + y^{2} - 2 λ x = 0$ from any point $(α, β)$ .

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Solution

Let $L (h, k)$ be the foot of normal $C L$ which is drawn from the point $(α, β)$ . Its equation is
$y - k = \frac{k}{h - λ} (x - h)$
It passes through the point $(α, β)$
$∴ (β - k) (h - λ) = k (α - h)$
or $h - \frac{k (α - h)}{β - k} = λ$ or $\frac{h β - k α}{β - k} = λ . . . . (1)$
But $(h, k)$ lies on the circle $h^{2} + k^{2} = 2 λ h . . . . (2)$
Eliminating the variable $λ$ from $(1)$ and $(2)$ , we get
$\frac{h β - k α}{β - k} = \frac{h^{2} + k^{2}}{2 h}$
$∴$ Locus of $(h, k)$ is
$2 x (x β - y α) = (x^{2} + y^{2}) (β - y)$ .

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