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`

Options

A. circle
B. parabola
C. ellipse
D. hyperbola

Explanation

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)`

`:.x_(1)=h` and `(y_(1))/(2)=kimpliesx_(1)=h` and `y_(1)=2k`
But, `(x_(1),y_(1))` lies on `xy=c^(2)`
`:.h.(2,k)=c^(2)implieshk=c^(2)//2`
Hence, the locus of `(h,k)` is `xy=c^(2)//2`, which is a hyperbola.