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)

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