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If P N is the perpendicular from a point...

If `P N` is the perpendicular from a point on a rectangular hyperbola `x y=c^2` to its asymptotes, then find the locus of the midpoint of `P N`

A

Circle

B

Parabola

C

Ellipse

D

Hyperbola

Text Solution

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The correct Answer is:
D

Let `xy=c^(2)` be the rectangular hyperbola, and let `P(x_(1),y_(1))` be a point on it. . Let Q (h,k) be the mid-point of PN.
Then the coordinates of Q are `(x_(1),(y_(1))/(2))`
`therefore x_(1) = h` and `(y_(1))/(2)=k`
`implies x_(1)=h` and `y_(1) =2k`
But `(x_(1),y_(1))` lies on `xy = c^(2)`
`h.(2k) = c^(2) implies hk = c^(2)//2.`
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