Question

If $t$ is a parameter, then $x=a(t+\frac{1}{t})$ and $y=b(t-\frac{1}{t})$ represent

• A. An ellipse
• B. A circle
• C. A pair of straight lines
• D. A hyperbola

Answer 1

The correct option is D A hyperbola

Given equations $x=a(t+\frac{1}{t})$ and $y=b(t-\frac{1}{t})$ can be rewritten as

$\frac{x}{a}=t+\frac{1}{t}$ and $\frac{y}{b}=t-\frac{1}{t}$

On squaring and subtracting, we get

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

$=(t^2+\frac{1}{t^2}+2)-(t^2+\frac{1}{t^2}-2)$

$=t^2+\frac{1}{t^2}+2-t^2-\frac{1}{t^2}+2=4$

$\Rightarrow \frac{x^2}{a^2}-\frac{y^2}{b^2}=4$

$\Rightarrow \frac{x^2}{4a^2}-\frac{y^2}{4b^2}=1$ which represents a hyperbola