<p>Let the length of common chord be 2<span>a</span>. If two circles satisfy \[\sqrt{9 - a^2} + \sqrt{16 - a^2} = 5\] and the length of common chord is \[\frac{2a}{5} = \frac{k}{5}\], find <span>k</span>.</p>
Step-by-Step Solution
Key Concept: Solve the constraint equation for the semi-chord length a, then extract the numerator k from the expression for the full chord length.
<p><strong>Step 1:</strong> Given equation: \[\sqrt{16 - a^2} = 5 - \sqrt{9 - a^2}\]</p><p><strong>Step 2:</strong> Square both sides: \[16 - a^2 = 25 + 9 - a^2 - 10\sqrt{9 - a^2}\]</p><p><strong>Step 3:</strong> Simplify: \[16 = 34 - 10\sqrt{9 - a^2}\]</p><p>\[10\sqrt{9 - a^2} = 18\]</p><p><strong>Step 4:</strong> Square again: \[100(9 - a^2) = 324\]</p><p>\[900 - 100a^2 = 324\]</p><p>\[100a^2 = 576\]</p><p>\[a = \frac{24}{10}\]</p><p><strong>Step 5:</strong> Length of common chord = \[2a = \frac{24}{5}\]</p><p>Therefore, <span>k</span> = 24</p>
Correct Answer: 24