Question

Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$. Then $S=\left\{x \in R : \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$ :

• A. is an empty set
• B. contains exactly one element
• C. contains exactly two elements
• D. is an infinite set

Answer 1

Answer: (C)

$ \left(x+\frac{1}{2}\right)^2=\left(y+\frac{1}{4}\right)$
$ y=\left(x^2+x\right) $
$ \tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^2+x+1}=\pi / 2$
$ 0 \leq x^2+x+1 \leq 1$
$x^2+x \leq 0 $....(1)
$ \text { Also } x^2+x \geq 0 $....(2)
$ \therefore x^2+x=0 \Rightarrow x=0,-1$
$S$ contains 2 element.