Question:
<p>a, b > 0. The length of the common chord of the circles (x - a)<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup> and x<sup>2</sup> + (y - b)<sup>2</sup> = b<sup>2</sup> is</p>
<p style="display:inline"><span class="math-tex">\(\sqrt{a+b}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{a+b}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2 a b}{\sqrt{a^{2}+b^{2}}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2 a b}{a+b}\)</span></p>
Step-by-Step Solution
Key Concept: The common chord's length is twice the altitude to the hypotenuse of the right triangle formed by the origin and the centers of the two circles.
<html><body><p><img alt="" data-imgur-src="IWirDEY.png" height="130" src="https://media-mycbseguide.s3.amazonaws.com/images/imgur/1623761585-5zaxcu.jpg" width="116"/><br/>
Let |OM| = l<br/>
Express area of triangle OPQ in two ways and equate<br/>
<span class="math-tex">\(\frac{1}{2} a b=\frac{1}{2} l \sqrt{a^{2}+b^{2}}\)</span><br/>
<span class="math-tex">\(\Rightarrow l=\frac{a b}{\sqrt{a^{2}+b^{2}}}\)</span><br/>
Length of common chord = 2l <span class="math-tex">\(=\frac{2 a b}{\sqrt{a^{2}+b^{2}}}\)</span></p></body></html>
Correct Answer: C