Question:
<p>Let the tangent to the parabola <span class="math-tex">\(S: y^{2}=2 x\)</span> at the point <span class="math-tex">\({P}(2,2)\)</span> meet the <span class="math-tex">\(x\)</span>-axis at Q and normal at it meet the parabola <span class="math-tex">\(S\)</span> at the point R. Then, the area (in sq. units) of the triangle PQR is equal to:</p>
<p style="display:inline"><span class="math-tex">\(\frac{15}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{35}{2}\)</span></p>
<p style="display:inline">25</p>
<p style="display:inline"><span class="math-tex">\(\frac{25}{2}\)</span></p>
Step-by-Step Solution
Key Concept: Determine the coordinates of the triangle's vertices by deriving the tangent and normal equations at point P, then apply the determinant formula for the area.
<p><img src="https://media-mycbseguide.s3.amazonaws.com/images/question_images/1756969115-cr94hs.jpg" style="height:140px; width:200px" /><br />
Tangent at <span class="math-tex">${P}: y y_{1}=2 {a}\left(x+x_{1}\right)$</span><br />
<span class="math-tex">$y(2)=2\left(\frac{1}{2}\right)(x+2)$</span><br />
<span class="math-tex">$\Rightarrow 2 y=x+2$</span><br />
<span class="math-tex">$\therefore {Q}=(-2,0)$</span><br />
slope of tangent <span class="math-tex">${P}=\frac{1}{2}$</span><br />
Normal at <span class="math-tex">${P}: y-2=-\frac{1}{\left(\frac{1}{2}\right)}(x-2)$</span><br />
<span class="math-tex">$\Rightarrow y-2=-2(x-2) \left[\because m_{1} m_{2}=-1\right]$</span><br />
<span class="math-tex">$\Rightarrow y=6-2 x$</span><br />
<span class="math-tex">$\therefore$</span> Now, solving with <span class="math-tex">$y^{2}=2 x \Rightarrow {R}\left(\frac{9}{2},-3\right)$</span><br />
<span class="math-tex">$\therefore \operatorname{Area}(\triangle {PQR})=\frac{1}{2}\left|\begin{array}{ccc}2 & 2 & 1 \\ -2 & 0 & 1 \\ \frac{9}{2} & -3 & 1\end{array}\right|$</span><br />
<span class="math-tex">$=\frac{1}{2}\left|2(0+3)-2\left(-2-\frac{9}{2}\right)+1(+6-0)\right|$</span><br />
<span class="math-tex">$=\frac{25}{2}$</span> sq. units</p>
Correct Answer: D