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

(\frac{a}{2}, 0)

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(0, \frac{a}{2})

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(0, a)

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(a, 0)

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Solution

The correct options are
A $(\frac{a}{2}, 0)$
B $(0, a)$
Let $P (h, k)$ be the point of intersection of the tangents at the extremities of the chord $A B$ of the circle $x^{2} + y^{2} = a^{2}$
Since $A B$ 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 circle $x^{2} + y^{2} - 2 a x = 0,$
then $\frac{h \cdot a + k \cdot 0 - a^{2}}{\sqrt{h^{2} + k^{2}}} = \pm a$
$\Rightarrow (h - a)^{2} = h^{2} + k^{2}$
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 $(c)$
Ans: A,C

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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___

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The locus of the point of intersection of the tangents at the extremities of a chord of the circlex2+y2=a2 which touches the circle 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 circle $x^{2} + y^{2} - 2 a x = 0$ passes through the point