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Question

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+y22ax=0 passes through the point___


A

(a/2, 0)

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B

(0, a/2)

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C

(0, a)

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D

(a, 0)

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Solution

The correct options are
A

(a/2, 0)


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 x2+y2=a2. Since AB is the chord of contact of the tangents from P to this circle, its equation is hx+ky=a2. If this line touches the circles x2+y22ax=0, then
h.a+k.0a2h2+k2=±a(ha)2=h2+k2
Therefore, the locus of (h, k) is (xa)2=x2+y2, or y2=a(a2x), which passes through the points given in (a) and (c).


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