The locus of the point of intersection of the | vedclass

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Question

(A) Let the point of intersection of the tangents be $P(h, k)$.The equation of the chord of contact of the tangents from $P(h, k)$ to the circle $x^2

Vedclass · Accepted Answer

$y^2 = a(a - 2x)$

Vedclass · Answer

(A) Let the point of intersection of the tangents be $P(h, k)$.The equation of the chord of contact of the tangents from $P(h, k)$ to the circle $x^2 + y^2 = a^2$ is given by $hx + ky = a^2$,or $hx + ky - a^2 = 0$.This chord touches the circle $x^2 + y^2 - 2ax = 0$,which has center $(a, 0)$ and radius $a$.The perpendicular distance from the center $(a, 0)$ to the line $hx + ky - a^2 = 0$ must be equal to the radius $a$:$\frac{|h(a) + k(0) - a^2|}{\sqrt{h^2 + k^2}} = a$$|ah - a^2| = a\sqrt{h^2 + k^2}$$|h - a| = \sqrt{h^2 + k^2}$Squaring both sides:$(h - a)^2 = h^2 + k^2$$h^2 - 2ah + a^2 = h^2 + k^2$$k^2 = a^2 - 2ah = a(a - 2h)$Replacing $(h, k)$ with $(x, y)$,the locus is $y^2 = a(a - 2x)$.

The locus of the point of intersection of the tangents at the extremities of a chord of the circle $x^2 + y^2 = a^2$ which touches the circle $x^2 + y^2 = 2ax$ is

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