<p>The normals drawn at P(t<sub>1</sub>) and Q(t<sub>2</sub>) to y<sup>2</sup> = 4ax meet the curve again at a point. The tangents at P and Q meet at R. As P and Q vary, the circumcentre of <span class="math-tex">\(\Delta\)</span>PQR describes</p>
<p style="display:inline">a parabola</p>
<p style="display:inline">a circle</p>
<p style="display:inline">a line parallel to the y - axis</p>
<p style="display:inline">a line parallel to the x - axis</p>
Step-by-Step Solution
Key Concept: The circumcircle of triangle PQR is the circle with the intersection of tangents (R) and the intersection of normals (S) as its diameter because the tangent and normal at any point on a parabola are mutually perpendicular.
<html><body><p><img alt="" data-imgur-src="udixsdN.png" height="158" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1657625458-nb9pxa.jpg" width="133"/><br/>
P = (<span class="math-tex">$a t_{1}^{2}$</span>, 2at<sub>1</sub>), Q = (<span class="math-tex">$a t_{2}^{2}$</span>, 2at<sub>2</sub>)<br/>
Then R = (at<sub>1</sub>t<sub>2</sub>, a(t<sub>1</sub> + t<sub>2</sub>))<br/>
The normals drawn at P and Q meet the curve at S. Let S = (<span class="math-tex">$a t_{3}^{2}$</span>, 2at<sub>3</sub>),<br/>
Then t<sub>3</sub> = -t - <span class="math-tex">$\frac{2}{t}$</span> where t = t<sub>1</sub> or t<sub>2</sub><br/>
<span class="math-tex">$\Leftrightarrow$</span> t<sup>2</sup> + t.t<sub>3</sub> + 2 = 0<br/>
<span class="math-tex">$\Rightarrow$</span> t<sub>1</sub> + t<sub>2</sub> = -t<sub>3</sub> and t<sub>1</sub>t<sub>2</sub> = 2<br/>
<span class="math-tex">$\Rightarrow$</span> R = (2a, -at<sub>3</sub>)<br/>
Clearly <span class="math-tex">$\square$</span>PROS is a cyclic quadrilateral,<br/>
<span class="math-tex">$\angle$</span>RPS = <span class="math-tex">$\angle$</span>RQS = 90°<br/>
<span class="math-tex">$\Rightarrow$</span> RS is the diameter of the circumcircle of <span class="math-tex">$\square$</span>PRQS (This circle is same as the circumcircle of <span class="math-tex">$\triangle$</span>PQR.)<br/>
Let the circumcentre be C(h, k).<br/>
<span class="math-tex">$\Rightarrow$</span> Circumcentre C = Midpoint of RS<br/>
<span class="math-tex">$\Leftrightarrow(h, k)=\left(\frac{2 a+a t_{3}^{2}}{2}, \frac{-a t_{3}+2 a t_{3}}{2}\right)$</span><br/>
<span class="math-tex">$\Rightarrow$</span> 2h = 2a + <span class="math-tex">$a t_{3}^{2}$</span>, 2k = at<sub>3</sub><br/>
<span class="math-tex">$\Rightarrow$</span> 2h = 2a + <span class="math-tex">$a\left(\frac{2 k}{a}\right)^{2}$</span><br/>
<span class="math-tex">$\Leftrightarrow$</span> 2k<sup>2</sup> = a(h - a)<br/>
<span class="math-tex">$\Rightarrow$</span> The circumcentre C moves on a parabola given by 2y<sup>2</sup> = a(x - a)</p></body></html>
Correct Answer: A