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Question

Prove that the product of perpendiculars from any point on the hyperbola x2a2y2b2=1 to its asymptotes is constant and the value is a2b2a2+b2

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Solution

The equation of asymptotes are bx+ay=0 and bxay=0
Let the point on hyperbola be (p,q)
Perpendicular distance from the point (p,q) to the line bx+ay=0 is bp+aya2+b2
Perpendicular distance from point (p,q) to the line bxay=0 is bpaqa2+b2
The product of distances will be b2p2a2q2a2+b2
The point (p,q) lie on hyperbola , so by substituting we get b2p2a2q2=a2b2
Therefore the product of distances is a2b2a2+b2
Hence proved

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