Question

If x = a cos θ and y = b cot θ, show that:

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

Answer 1

${\displaystyle \frac{a^2}{x^2}-\frac{b^2}{y^2}}$

= ${\displaystyle \frac{a^2}{a^2{\cos }^2\theta }-\frac{b^2}{b^2{\cot }^2\theta }}$

= ${\displaystyle \frac{1}{{\cos }^2\theta }-\frac{{\sin }^2\theta }{{\cos }^2\theta }}$

= ${\displaystyle \frac{1-{\sin }^2\theta }{{\cos }^2\theta }}$

= ${\displaystyle \frac{{\cos }^2\theta }{{\cos }^2\theta }}$

= 1