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Question

In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and X'Y' at B. Then AOB is _____.

A
80
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B
60
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C
85
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D
90
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Solution

The correct option is D 90

In quadrilateral APQB, we have

APO=90 and BQO=90
[ Tangent is perpendicular to the radius at the point of contact- By Theorem].

Now, APO+BQO+QBC+PAC=360
PAC+QBC=360(APO+BQO)=180 . . . (i)

We have

CAO=12PAC and CBO=12QBC [ Tangents from an external point are equally inclined to the line segment joining the centre to that point- By Theorem].

On adding CAO and CBO and using (i) we get,

CAO+CBO=12(PAC+QBC)=12×180=90 . . .(ii)

In Δ AOB, we have

CAO+AOB+CBO=180
AOB=180(CAO+CBO)=90 [using (ii)].

Hence, AOB=90


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