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Question

If the chord xcosα+ysinα=p of the hyperbola x216y218=1 subtends a right angle at the centre, and the diameter of the circle, concentric with the hyperbola to which the given chord is a tangent is d units, then the value of d4 is

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Solution

Equation of hyperbola is x216y218=1
or 9x28y2144=0
Homogenizing hyperbola equation using
xcosα+ysinαp=1
we get
9x28y2144(xcosα+ysinαp)2=0
Since these lines are perpendicular to each other
9p28p2144(cos2α+sin2α)=0
p2=144 or p=± 12
radius=|p|cos2α+sin2α=12
radius of the circle =12 unit
and, diameter of the circle (d)=24 unit
d4=6

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