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Question

Two tangents are drawn from a point P to the circle x2+y22x4y+4=0, such that the angle between these tangents is tan1(125), where tan1(125)(0,π). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of ΔPAB and ΔCAB is:

A
11:4
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B
9:4
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C
2:1
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D
3:1
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Solution

The correct option is B 9:4
Given : x2+y22x4y+4=0
r=1+44=1tanθ=125


Respective areas are
ΔPAB=12×L2sinθΔCAB=12×r2sin(πθ)ΔPABΔCAB=(Lr)2ΔPABΔCAB=cot2θ2ΔPABΔCAB=2cos2θ22sin2θ2ΔPABΔCAB=1+cosθ1cosθΔPABΔCAB=1+5131513=94

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