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 circles $x^{2} + y^{2} - 2 a x = 0$ passes through the point___

(a/2, 0)

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(0, a/2)

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(0, a)

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(a, 0)

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Solution

The correct option is C (0, a)
Let P(h, k) be the point of intersection of the tangents at the extremities of the chord AB of the circles $x^{2} + y^{2} = a^{2} .$ Since AB is the chord of contact of the tangents from P to this circle, its equation is $h x + k y = a^{2} .$ If this line touches the circles $x^{2} + y^{2} - 2 a x = 0,$ then
$\frac{h . a + k .0 - a^{2}}{\sqrt{h^{2} + k^{2}}} = \pm a \Rightarrow (h - a)^{2} = h^{2} + k^{2}$
Therefore, the locus of (h, k) is $(x - a)^{2} = x^{2} + y^{2}$ , or $y^{2} = a (a - 2 x),$ which passes through the points given in (a) and (c).

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The locus of the point of intersection of the tangents at the extremities of a chord of the circle x2+y2=a2 which touches the circles x2+y2−2ax=0 passes through the point___

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 circles $x^{2} + y^{2} - 2 a x = 0$ passes through the point___