Let [y=f(x)] represent a parabola with focus [(-(1)/(2), 0)] and directrix [y=-(1)/(2)].

Question

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then

$$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$$ :

Accepted Answer

$\left(x+\frac{1}{2}ight)^{2}=\left(y+\frac{1}{4}ight)$

$y=\left(x^{2}+xight)$

$	an ^{-1} \sqrt{\mathrm{x}(\mathrm{x}+1)}+\sin ^{-1} \sqrt{\mathrm{x}^{2}+\mathrm{x}+1}=\pi / 2$

$0 \leq \mathrm{x}^{2}+\mathrm{x}+1 \leq 1$

$x^{2}+x \leq 0$

Also $x^{2}+x \geq 0$

$	herefore \mathrm{x}^{2}+\mathrm{x}=0 \Rightarrow \mathrm{x}=0,-1$

$\mathrm{S}$ contains 2 element.

Solutions · 1