Question

The equations $x=\frac{e^t+e^{-t}}{2},y=\frac{e^t-e^{-t}}{2}$ where $t$ is real number, represents.

• A. An ellipse
• B. A parabola
• C. A hyperbola
• D. A circle

Answer 1

Answer: (C)

A hyperbola

$x=\frac{e^t+e^{-t}}{2}$ $y=\frac{e^{+}-e^{-t}}{2}$

$x^2=\frac{e^{2t}+e^{-2t}+{2e}^t{e}^{-t}}{4}=\frac{e^{2t}+e^{-2t}+2}{4}$

$y^2=\frac{e^{2t}+e^{-2t}{-2e}^t{e}^{-t}}{4}=\frac{e^{2t}+e^{-2t}-2}{4}$

$x^2-y^2=\frac{e^{2t}+e^{-2t}+2-e^{2t}-e^{-2t}+2}{4}$

$=\frac{4}{4}=1$

$x^2-y^2=1$

$\therefore$ It is a Hyperbola with eccentricity $\sqrt{2}$.