Question:
<p>A circle touches the parabola y<sup>2</sup> = 4x at M (1, 2) and also touches its directrix. The y-coordinate of the point of contact of the circle and the directrix is:</p>
<p style="display:inline">2<span class="math-tex">\(\sqrt 2\)</span></p>
<p style="display:inline">4</p>
<p style="display:inline">2</p>
<p style="display:inline"><span class="math-tex">\(\sqrt 2\)</span></p>
Step-by-Step Solution
Key Concept: For a circle tangent to parabola y²=4x at point M(1,2), the center lies on the normal at M. Since the circle also touches the directrix x=-1, the distance from center to directrix equals the radius, and the center is equidistant from M and the directrix.
<html><body><p><img alt="" data-imgur-src="ZjUFbFX.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1619691751-r6x6bh.jpg" style="width: 150px; height: 122px;"/><br/>
y<sup>2</sup> = 4x<br/>
2y <span class="math-tex">$\frac{{dy}}{{dx}}$</span> = 4<br/>
<span class="math-tex">$m_T=\frac{2}{\mathrm{y}}=\frac{2}{2}$</span> = 1<br/>
Circle <span class="math-tex">$\rightarrow$</span> S + <span class="math-tex">$\lambda$</span>L = 0<br/>
(x + 1)<sup>2</sup> + (y + <span class="math-tex">$\alpha$</span>)<sup>2 </sup>+ <span class="math-tex">$\lambda$</span>(x + 1) = 0 ...(i)<br/>
differentiate,<br/>
2(x + 1) + 2(y - <span class="math-tex">$\alpha$</span>) <span class="math-tex">$\frac{{dy}}{{dx}}+\lambda$</span> = 0<br/>
x = 1, y = 2<br/>
4 + 2(2 - <span class="math-tex">$\alpha$</span>) m<sub>T</sub> + <span class="math-tex">$\lambda$</span> = 0<br/>
<span class="math-tex">$\lambda$</span> = 2<span class="math-tex">$\alpha$</span> - 8 ...(ii)<br/>
(1, 2) satisfies eq. (i)<br/>
2 = (2 - <span class="math-tex">$\alpha$</span>)<sup>2</sup> + 2<span class="math-tex">$\lambda$</span> = 0<br/>
<span class="math-tex">$\alpha$</span><sup>2</sup> - 4<span class="math-tex">$\alpha$</span> + 8 + 2(2<span class="math-tex">$\alpha$</span> - 8) = 0<br/>
<span class="math-tex">$\alpha$</span><sup>2</sup> = 8<br/>
<span class="math-tex">$\alpha$</span> = 2 <span class="math-tex">$\sqrt 2$</span></p></body></html>
Correct Answer: A