If the straight line `xcosalpha+ysinalpha=p` touches the curve `(x^2)/(a^2)-(y^2)/(b^2)=1,` then prove that `a^2cos^2alpha-b^2sin^2alpha=p^2dot`

Explanation

Tangent to hyperbola at `(x_1,y_1)`

`(x x_1)/a^2-(yy_1)/b^2=1`

`(xcosalpha)/b+(ysinalpha)/b=1`

`x_1/a=cosalpha/p,x_1=(acosalpha)/p`

`-y_1/b=sinalpha/p,y_1=(-bsinalpha)/p`

`(a^2cos^2alpha)/p-(b^2sin^2alpha)/p=p`

`a^2cos^2alpha-b^2sin^2alpha=p^2`.