The circle x2+y2−8x=0 and hyperbola x29−y24=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±√5y−4=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 x29−y24=1 is y=mx±√9m2−4
It is a tangent to x2+y2−8x=0 also ∴4m±√9m2−4√1+m2=4 ⇒4m±√9m2−4=4√1+m2
Squaring both sides, we get 16m2±8m√9m2−4+9m2−4=16(1+m2) ⇒9m2−20=±8m√9m2−4
Again squaring both sieds, we get ⇒495m4+104m2−400=0 ⇒m2=45,−10099 ∴m=±2√5
Hence possible diagram will be
∴ Equations of common tangents are y=2√5x+4√5,y=−2√5x−4√5 ⇒2x±√5y+4=0
and intersection point of tangents will be (−2,0)