Question

If $x=asec\theta cos\varphi y=bsec\theta sin\varphi andz=ctan\theta$, show that $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

Answer 1

LHS$=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}$

$=\left({\sec }^2\theta {\cos }^2\varphi \right)+\left({\sec }^2\theta {\sin }^2\varphi \right)-{\tan }^2\theta$

$={\sec }^2\theta ({\sin }^2\varphi +{\cos }^2\varphi )-{\tan }^2\theta$

$={\sec }^2\theta -{\tan }^2\theta$ ...... $[sin²\theta +cos²\theta =1]$

$=1$ ...... $[{\sec }^2\theta =1+{\tan }^2\theta ]$

$=RHS$

Hence proved