Question

If t is parameter, then the equations x = a $\left(t+\frac{1}{t}\right),y=b\left(t-\frac{1}{t}\right)$ represent __________________.

Answer 1

Given equations,

$$\begin{gathered} x=a\left(t+\frac{1}{t}\right)\mathrm{and}y=b\left(t-\frac{1}{t}\right) \\ \mathrm{can}\mathrm{be}\mathrm{written}\mathrm{as}, \\ \frac{x}{a}=t+\frac{1}{t}and\frac{y}{b}=t-\frac{1}{t} \\ \therefore {\left(\frac{x}{a}\right)}^2-{\left(\frac{4}{b}\right)}^2={\left(t+\frac{1}{t}\right)}^2-{\left(t-\frac{1}{t}\right)}^2 \\ =t^2+\frac{1}{t^2}+2-\left(t^2+\frac{1}{t^2}-2\right) \\ \mathrm{i}.\mathrm{e}.{\left(\frac{x}{a}\right)}^2-{\left(\frac{4}{b}\right)}^2=4 \\ \mathrm{i}.\mathrm{e}.\frac{x^2}{4a^2}-\frac{y^2}{4b^2}=1\mathrm{which}\mathrm{represents}\mathrm{a}\mathrm{hyperbola} \end{gathered}$$