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Question

Let the point B be the reflection of the point A(2,3) with respect to the line 8x6y23=0. Let ΓA and ΓB be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles ΓA and ΓB such that both the circles are on the same side of T. If C is the point of intersection of T and line passing through A and B, then the length of the line segment AC is

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Solution


Given line: 8x6y23=0
Distance of point A(2,3) from the given line
AR=∣ ∣8x16y12382+62∣ ∣ =∣ ∣8×26×32382+62∣ ∣ =2510=52

AR=52 and AB=5
From similar triangle
QBPA=CBCA12=BCAB+BC5+BC=2BCBC=5AC=5+5=10

Alternate Solution:
C is E.C.S (External Centre of Similitude), which divides ¯¯¯¯¯¯¯¯AB in the ratio r1:r2 externally.
ACBC=21
B is the midpoint of ¯¯¯¯¯¯¯¯AC

AC=4AR
=4×∣ ∣8x16y12382+62∣ ∣ =4×∣ ∣8×26×32382+62∣ ∣ =4×2510=10

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