Question:
<p>A line L is common tangent to the circle x<sup>2</sup> + y<sup>2</sup> = 1 and the parabola y<sup>2</sup> = 4x. If <span class="math-tex">\(\theta\)</span> is the angle which it makes with the positive x-axis, then tan<sup>2</sup> <span class="math-tex">\(\theta\)</span> is. equal to:</p>
<p style="display:inline">2 cos 36°</p>
<p style="display:inline">2 sin 15°</p>
<p style="display:inline">cos 36°</p>
<p style="display:inline">2 sin 18°</p>
Step-by-Step Solution
Key Concept: To find a common tangent, equate the slope-form equation of the tangent of the parabola to the tangency condition of the circle where the perpendicular distance from the center equals the radius.
<html><body><p><img alt="" data-imgur-src="WLPRhRc.png" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1619691748-7ydz9x.jpg" style="width: 150px; height: 105px;"/><br/>
Tangent to the parabola<br/>
y<sup>2</sup> = 4x is y = mx + <span class="math-tex">\(\frac 1m\)</span> ...(i)<br/>
m<sup>2</sup>x - my + 1 = 0<br/>
As, it touches the circle x<sup>2</sup> + y<sup>2</sup> = 1, so<br/>
<span class="math-tex">\(\left|\frac{{1}}{\sqrt{{m}^{4}+{m}^{2}}}\right|={1} \Rightarrow\)</span> m<sup>4</sup> + m<sup>2</sup> - 1 = 0<br/>
<span class="math-tex">\(\therefore\)</span> m<sup>2</sup> = tan<sup>2</sup> <span class="math-tex">\(\theta=\frac{-1 \ \pm \ \sqrt{1 \ + \ 4}}{2}\)</span><br/>
= <span class="math-tex">\(\frac{\sqrt{5} \ - \ 1}{2}=2\left(\frac{\sqrt{5} \ - \ 1}{4}\right)\)</span> = 2 sin 18°</p></body></html>
Correct Answer: D