The product of the perpendiculars from the foci on any tangent to the hyperbola x264−y29=1 is

Equation of a tangent at  (8sec⁡ θ, 3tan⁡ θ) the hyperbola is x8sec⁡ θ−y3tan⁡ θ=1If e is the eccentricity of the hyperbola; product of the perpendiculars=(esec⁡θ−1)(−esec⁡θ−1)sec2⁡θ64+tan2⁡θ9=1−e2sec2⁡θ×(64×9)9sec2⁡θ+64tan2⁡θ=1−64−964sec2⁡θ(64×9)9sec2⁡θ+64tan2⁡θ=64tan2⁡θ+9sec2⁡θ9sec2⁡θ+64tan2⁡θ×9=9.