If `a x+b y=1` is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , then `a^2-b^2` is equal to (a)`1/(a^2e^2)` (b) `a^2e^2` (c)`b^2e^2` (d) none of these

Options

A. `1//a^(2)e^(2)`
B. `a^(2)e^(2)`
C. `b^(2)e^(2)` none of these
D. none of these

Answer

A

Explanation

Any tangent to hyperbola is
`(x)/(a) sec theta-(y)/(b) tan theta=1" (1)"`
The given tangent is
`ax+by=1" (2)"`
Comparing (1) and (2), we have
`sec theta=a^(2)and tan theta=-b^(2)`
Eliminating `theta`, we have
`a^(4)-b^(4)=1`
`"or "(a^(2)-b^(2))(a^(2)+b^(2))=1`
Also, `a^(2)+b^(2)=a^(2)e^(2)`
`"or "a^(2)-b^(2)=(1)/(a^(2)e^(2))`