Question

Let $C$ be a conic whose parametric equations are $x=\frac{e^t+e^{-t}}{2}$ and $y=\frac{e^t-e^{-t}}{3};$ where $t$ is a parameter. Then which of the following is/are true?

• A. $C$ is a hyperbola with eccentricity $\frac{\sqrt{13}}{3}$
• B. $C$ is a hyperbola with eccentricity $\frac{\sqrt{5}}{3}$
• C. $C$ intersects $y-$ axis at two points
• D. Auxilary circle of $C$ passes through the point $(\frac{1}{2},-\frac{\sqrt{3}}{2})$

Answer 1

Answer: (A), (D)

The correct options are
A $C$ is a hyperbola with eccentricity $\frac{\sqrt{13}}{3}$
D Auxilary circle of $C$ passes through the point $(\frac{1}{2},-\frac{\sqrt{3}}{2})$

Given, $x=\frac{e^t+e^{-t}}{2}$
$\Rightarrow 2x=e^t+e^{-t}\dots \left(1\right)$

and $y=\frac{e^t-e^{-t}}{3}$
$\Rightarrow 3y=e^t-e^{-t}\dots \left(2\right)$

From $(1{)}^2-(2{)}^2,$ we get
$4x^2-9y^2=4$
$\Rightarrow \frac{x^2}{1}-\frac{y^2}{(2/3{)}^2}=1$
which is the equation of hyperbola with $a=1,b=\frac{2}{3}.$
$\therefore e=\sqrt{1+\frac{b^2}{a^2}}=\frac{\sqrt{13}}{3}$

When $x=0$
$\Rightarrow y^2=-\frac{4}{9}$
So, given hyperbola does not intersect the $y-$axis.

Equation of auxilary circle of the hyperbola is
$x^2+y^2=a^2\Rightarrow x^2+y^2=1$
which passes through the point $(\frac{1}{2},-\frac{\sqrt{3}}{2}).$