Question:
<p>Tangents drawn from the point (4, 3) to the circle x<sup>2</sup> + y<sup>2</sup> - 2x - 4y = 0 are inclined at an angle</p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{6}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi}{3}\)</span></p>
Step-by-Step Solution
Key Concept: Determine the angle between tangents by using the radius and tangent length to find the half-angle in the right-angled triangle formed with the circle's center.
<html><body><p><img alt="" data-imgur-src="04sIkpb.png" height="127" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761546-edwjky.jpg" width="157"/><br/>
Centre of the circle C = (1, 2)<br/>
Radius = |CT| = <span class="math-tex">$\sqrt{5}$</span><br/>
Let P = (4, 3)<br/>
<span class="math-tex">$|\mathrm{PC}|=\sqrt{(4-1)^{2}+(3-2)^{2}}=\sqrt{10}$</span><br/>
<span class="math-tex">$\Rightarrow|\mathrm{PT}|=\sqrt{10-5}=\sqrt{5}$</span><br/>
<span class="math-tex">$\Rightarrow \triangle$</span>PCT is an isosceles triangle<br/>
<span class="math-tex">$\Rightarrow$</span> Angle between tangents from P = <span class="math-tex">$\frac{\pi}{2}$</span></p></body></html>
Correct Answer: A