Question

Assertion :The point $P[\frac{a}{2}(t+\frac{1}{t}),\frac{b}{2}(t-\frac{1}{t})]$ lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ for infinite values of $t$. Reason: Locus of point $P$ is hyperbola if $t\in R$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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