Question

Eliminate $\theta$, if$x=a\sec \theta ,y=b\tan \theta$

Answer 1

Solution:-

$x=a\sec \theta \Rightarrow \sec \theta =\frac{x}{a}$

$y=b\tan \theta \Rightarrow \tan \theta =\frac{y}{b}$

As we know that,

${\sec }^2\theta -{\tan }^2\theta =1$

$\Rightarrow {\left(\frac{x}{a}\right)}^2-{\left(\frac{y}{b}\right)}^2=1$

$\Rightarrow \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\Rightarrow \frac{b^2{x}^2-a^2{y}^2}{a^2{b}^2}=1$

$\Rightarrow b^2{x}^2-a^2{y}^2=a^2{b}^2$

$\Rightarrow b^2{x}^2=a^2{b}^2+a^2{y}^2$

$\Rightarrow x^2=\frac{a^2(b^2+y^2)}{b^2}$

Therefore, $x^2=\frac{a^2(b^2+y^2)}{b^2}$

which is the required $\theta$-eliminate.