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

Equation of parabola with focus $( - \dfrac{1}{2}, 0 )$ and directrix $y = - \dfrac{1}{2}$ is:

${\left(x + \dfrac{1}{2}\right)}^{2} = \left(y + \dfrac{1}{4}\right)$

$⇒ y = \left({x}^{2} + x\right)$

Put $f ( x ) = y = x + {x}^{2} in$S$,</p><p>$\tan^{- 1}\sqrt{x\left(x + 1\right)} + \sin^{- 1}\sqrt{{x}^{2} + x + 1} = \dfrac{\pi}{2}$</p><p>$⇒ 0 \le {x}^{2} + x + 1 \le 1$</p><p>$⇒ {x}^{2} + x \le 0 \ldots \left(1\right)$</p><p>Also,</p><p>${x}^{2} + x \ge 0 \ldots \left(2\right)$</p><p>By \left(1\right) \text{ and } \left(2\right)$, we have

${x}^{2} + x = 0 ⇒ x = 0, - 1$

$S$ contains $2$ element.