Question

The point $\{\frac{a}{2}(t+\frac{1}{t}),\frac{b}{2}(t-\frac{1}{t})\}$ lies on which of the following hyperbolas for all values of $t(t\ne 0)$

• A. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
• B. $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$
• C. $\frac{x^2}{4a^2}-\frac{y^2}{4b^2}=1$
• D. none of these

Answer 1

The correct option is A $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Let $P(h,k)$ be the point.
Given $h=\frac{a}{2}(t+\frac{1}{t})..\left(1\right)$ and $k=\frac{b}{2}(t-\frac{1}{t})..\left(2\right)$
Using (1) and (2)
$\frac{h}{a}+\frac{k}{b}=t$ and $\frac{h}{a}-\frac{k}{b}=\frac{1}{t}$
Eliminating parameter $t$ we get
$\frac{h^2}{a^2}-\frac{k^2}{b^2}=1$
Thus locus of $P(h,k)$ is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, which is a hyperbola
Hence, option 'A' is correct