Find the locus of the mid points of the chords of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` which subtent a right angle at the origin.

Options

A. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=1`
B. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))+(1)/(b^(2))`
C. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))`
D. `(x^(2))/(a^(4))-(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))`

Explanation

Let `P(h,k)` be the pole of a chord of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. Then, the equation of the polar is
`(hx)/(a^(2))-(ky)/(b^(2))=1`………`(i)`
The combined equation of the lines joining the origin to the points of intersection of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` and the line `(i)` is
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=((hx)/(a^(2))-(ky)/(b^(2)))^(2)`
Since the lines given by the above equation are at right angle.
`:. "Coefficient of " x^(2)+ "Cofficient of" y^(2)=0`
`implies(1)/(a^(2))-(h^(2))/(a^(4))-(1)/(b^(2))-(k^(2))/(b^(4))=0implies(h^(2))/(a^(4))+(k^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))`
Hence, the locus of `P(h,k)` is `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))`