Question:
<p>From any point on the circle x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup> tangents are drawn to the circle x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup>sin<sup>2</sup> <span class="math-tex">\(\alpha\)</span>, the angle between them is</p>
<p style="display:inline"><span class="math-tex">\(\alpha\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\alpha}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(2+\alpha\)</span></p>
<p style="display:inline"><span class="math-tex">\(2\alpha\)</span></p>
Step-by-Step Solution
Key Concept: For concentric circles, the angle between tangents is twice the angle in a right triangle where the outer radius is the hypotenuse and the inner radius is the opposite side.
<p><img alt="" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761551-vwpsdr.jpg" style="height:123px; width:132px" /><br />
Given two circles are concentric.<br />
Let <span class="math-tex">$\angle \mathrm{OPQ}=\theta$</span><br />
QP = a cos<span class="math-tex">$\alpha$</span><br />
tan <span class="math-tex">$\theta$</span> <span class="math-tex">$=\frac{a \sin \alpha}{a \cos \alpha}$</span> = tan <span class="math-tex">$\alpha$</span><br />
<span class="math-tex">$\Rightarrow \theta=\alpha$</span><br />
<span class="math-tex">$\Rightarrow$</span> Angle between tangents = <span class="math-tex">$\angle \mathrm{QPR}=2 \alpha$</span></p>
Correct Answer: D