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)

contains exactly two elements

Answer 2

(x+21​)2=(y+41​)
y=(x2+x)
tan−1x(x+1)​+sin−1x2+x+1​=π/2
0≤x2+x+1≤1
x2+x≤0....(1)
Also x2+x≥0....(2)
∴x2+x=0⇒x=0,−1
S contains 2 element.