Prove that the locus of the point of intersection of tangents at the extremities of a chord of the circle $x^{2} + y^{2} = a^{2}$ which touch the circle $x^{2} + y^{2} - 2 a x = 0$ is the parabola
$y^{2} = - 2 a (x - a / 2)$ .

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Solution

Let $P (h, k)$ be the point of intersection. Its chord of contact w.r.t. $x^{2} + y^{2} = a^{2}$ is $h x + k y - a^{2} = 0$ . It touches the circle $x^{2} + y^{2} - 2 a x = 0$ i.e. $C (a, 0), a$ .
Apply $p = r . ∴ \frac{a h - a^{2}}{\sqrt{(h^{2} + k^{2})}} = a$
Now square and generalize $h, k$ .

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Prove that the locus of the point of intersection of tangents at the extremities of a chord of the circle x2+y2=a2 which touch the circle x2+y2−2ax=0 is the parabolay2=−2a(x−a/2).

Let P(h,k) be the point of intersection. Its chord of contact w.r.t. x2+y2=a2 is hx+ky−a2=0. It touches the circle x2+y2−2ax=0 i.e. C(a,0),a.Apply p=r.∴ah−a2√(h2+k2)=aNow square and generalize h,k.

Prove that the locus of the point of intersection of tangents at the extremities of a chord of the circle $x^{2} + y^{2} = a^{2}$ which touch the circle $x^{2} + y^{2} - 2 a x = 0$ is the parabola
$y^{2} = - 2 a (x - a / 2)$ .

Let $P (h, k)$ be the point of intersection. Its chord of contact w.r.t. $x^{2} + y^{2} = a^{2}$ is $h x + k y - a^{2} = 0$ . It touches the circle $x^{2} + y^{2} - 2 a x = 0$ i.e. $C (a, 0), a$ .
Apply $p = r . ∴ \frac{a h - a^{2}}{\sqrt{(h^{2} + k^{2})}} = a$
Now square and generalize $h, k$ .