If the straight-line x cos α + y sin α = p touches the curve x2/a2 + y2/b2 = 1, then prove that a2 cos2 α + b2 sin2 α = p2.

The given bend is \[\mathbf{x2}/\mathbf{a2}\text{ }+\text{ }\mathbf{y2}/\mathbf{b2}\text{ }=\text{ }\mathbf{1}\] and the straight-line \[\mathbf{x}\text{ }\mathbf{cos}\text{ }\mathbf{\alpha }\text{ }+\text{ }\mathbf{y}\text{ }\mathbf{sin}\text{ }\mathbf{\alpha }\text{ }=\text{ }\mathbf{p}\]

Separating condition (I) w.r.t. x, we get

Accordingly, \[\mathbf{a2}\text{ }\mathbf{cos2}\text{ }\mathbf{\alpha }\text{ }+\text{ }\mathbf{b2}\text{ }\mathbf{sin2}\text{ }\mathbf{\alpha }\text{ }=\text{ }\mathbf{p2}.\]