Prove that the straight line `lx+my+n=0` touches the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` if `a^(2)l^(2)-b^(2)m^(2)=n^(2)`.

Options

A. `((a^(2)l)/(n),(b^(2)m)/(n))`
B. `(-(a^(2)l)/(n),(b^(2)m)/(n))`
C. `((a^(2)l)/(n),-(b^(2)m)/(n))`
D. `(-(a^(2)l)/(n),(-b^(2)m)/(n))`

Explanation

Let `(x_(1),y_(1))` be the pole of the line `lx+my+n=0` with respect to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. Then, the equation of the polar is
`(x x_(1))/(a^(2))-(y y_(1))/(b^(2))=1`………`(i)`
Since, `(x_(1),y_(1))` is the pole of the line `lx+my+n=0`. So, the polar of `(x_(1),y_(1))` is also the line
`lx+my+n=0`...`(ii)`
clearly, `(i)` and `(ii)` represent the same line. Therefore,
`(x_(1))/(a^(2)l)=(-y)/(b^(2)m)=(1)/(-n)impliesx_(1)=(-a^(2)l)/(n)`, `y_(1)=(b^(2)m)/(n)`.
Hence, the pole of the given line with respect to the given hyperbola is
`(-(a^(2)l)/(n),(b^(2)m)/(n))`