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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+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 x2+y2=a2 which touches the circles x2+y2−2ax=0 passes through the point
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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+y2−2ax=0, then
h.a+k.0−a2√h2+k2=±a⇒(h−a)2=h2+k2
Therefore, the locus of (h, k) is (x−a)2=x2+y2, or y2=a(a−2x), which passes through the points given in (a) and (c).
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+y2−2ax=0, then
h.a+k.0−a2√h2+k2=±a⇒(h−a)2=h2+k2
Therefore, the locus of (h, k) is (x−a)2=x2+y2, or y2=a(a−2x), which passes through the points given in (a) and (c).
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