Q. A general point on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is
A) $(a\sin{\theta,\ b\cos{\theta)\ (where\ \theta\ }}$is parameter)
B) $(a\tan{\theta,\ b\sec{\theta)\ (where\ \theta\ }}$is parameter)
C) $\left( a\frac{e^{t} + e^{- t}}{2},b\frac{e^{t} - e^{- t}}{2} \right)$ (where $t$ is parameter)
D) None of the above
Explanation
(c)
Now taking option (c).
Let $x = a\frac{e^{t} + e^{- t}}{2}\ \Rightarrow \frac{2x}{a} = e^{t} + e^{- t}$ …(i)
And $\frac{2y}{a} = e^{t} - e^{- t}$ …(ii)
On squaring and subtracting Eq. (ii) from Eq. (i), we get
$\frac{4x^{2}}{a^{2}} - \frac{4y^{2}}{b^{2}} = 4$
$\Rightarrow \ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.