Let the point B be the reflection of the point A 2 ,   3 with respect to the line 8 x - 6 y - 23 = 0 . Let T A and T B be circle of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circle T A and T B such that both the circle are on the same side of T . If C is

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Let the point B be the reflection of the point A 2 , &#160; 3 with respect to the line 8 x - 6 y - 23 = 0 . Let T A and T B be circle of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circle T A and T B such that both the circle are on the same side of T . If C is the point of intersection of T and the line passing through A and B , then the length of the line segment A C is ___________.

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Correct answer: A. A

Clearly, A C P and B C Q are similar triangles. Hence, A C B C = r 1 r 2 A C B C = 2 A C = 2 B C ...(i) Now, since B , is image of A in line 8 x - 6 y = 23 A D = B D A B = 2 A D ...(ii) A D = distance of A from line 8 x - 6 y = 23 A D = 16 - 16 - 23 10 = 5 2 Now as A C = 2 B C ⇒ A C = 2 ( A C - A B ) ⇒ A C = 2 A B ⇒ A C = 4 A D (from (ii)) A C = 4 × 5 2 A C = 10

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