Question:
<p>Let <span class="math-tex">\(y=f(x)\)</span> represent a parabola with focus <span class="math-tex">\(\left(-\frac{1}{2}, 0\right)\)</span> and directrix <span class="math-tex">\(y=-\frac{1}{2}\)</span>. Then <span class="math-tex">\({S}=\)</span> <span class="math-tex">\(\left\{x \in R: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}\left(\sqrt{f(x)+1)}=\frac{\pi}{2}\right\}:\right.\)</span></p>
<p style="display:inline">is an empty set</p>
<p style="display:inline">is an infinite set</p>
<p style="display:inline">contains exactly one element</p>
<p style="display:inline">contains exactly two elements</p>
Step-by-Step Solution
Key Concept: Determine the quadratic function from the parabola's geometric properties and solve the inverse trigonometric equation by identifying the intersection of the domains of the square root and inverse sine functions.
<p>Given focus of parabola <span class="math-tex">$\left(\frac{-1}{2}, 0\right)=(a, k+p)$</span> and directrix <span class="math-tex">$y=\frac{-1}{2}=k-p$</span><br />
<span class="math-tex">$\therefore$</span> Equation of parabola<br />
<span class="math-tex">$(x-a)^{2}=4 p(y-k)$</span><br />
<span class="math-tex">$k+p=0$</span><br />
<img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1756448030-ztcdmy.jpg" style="height:63px; width:60px" /><br />
<span class="math-tex">$\Rightarrow k=-\frac{1}{4}$</span><br />
<span class="math-tex">$p=\frac{1}{4}$</span><br />
<span class="math-tex">$\left(x+\frac{1}{2}\right)^{2}=4 \times \frac{1}{4}\left(y+\frac{1}{4}\right)$</span><br />
<span class="math-tex">$x^{2}+\frac{1}{4}+x=y+\frac{1}{4}$</span><br />
<span class="math-tex">$\therefore y=f(x)=x^{2}+x$</span> ...(1)<br />
<span class="math-tex">${S}=\left\{x \in {R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$</span><br />
<span class="math-tex">$\tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}$</span><br />
<span class="math-tex">$f(x) \geq 0$</span> and <span class="math-tex">$\sqrt{f(x)+1}$</span> cannot greater then 1.<br />
So <span class="math-tex">$f(x)$</span> must be 0<br />
i.e., <span class="math-tex">$f(x)=0$</span><br />
<span class="math-tex">$\Rightarrow x^{2}+x=0 \Rightarrow x(x+1)=0$</span><br />
<span class="math-tex">$\Rightarrow x=0,-1$</span><br />
<span class="math-tex">$\therefore {S}$</span> contain 2 elements.</p>
Correct Answer: D