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Question

The circle x2+y28x=0 and hyperbola x29y24=1 intersects at the points A and B. Then

A
equation of common tangents will be 2x±5y+4=0
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B
point of intersection of common tangents is (2,0)
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C
equation of common tangents will be 2x±5y4=0
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D
intersection point of common tangents will be (2,0)
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Solution

The correct option is B point of intersection of common tangents is (2,0)
A tangent to x29y24=1 is
y=mx±9m24

It is a tangent to x2+y28x=0 also
4m±9m241+m2=4
4m±9m24=41+m2
Squaring both sides, we get
16m2±8m9m24+9m24=16(1+m2)
9m220=±8m9m24
Again squaring both sieds, we get
495m4+104m2400=0
m2=45,10099
m=±25
Hence possible diagram will be


Equations of common tangents are
y=25x+45,y=25x45
2x±5y+4=0
and intersection point of tangents will be (2,0)

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