Circles
Intersecting Circles
Grade 11
Question:
<p>If the circle \(C_1: x^2 + y^2 = 16\) intersects another circle \(C_2\) of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to \(\frac{3}{4}\), then the coordinates of the centre of \(C_2\) are:</p>
<p>(a) \(\left(\pm\frac{9}{5}, \pm\frac{12}{5}\right)\)</p>
<p>(b) \(\left(\pm\frac{9}{5}, m\frac{12}{5}\right)\)</p>
<p>(c) \(\left(\pm\frac{12}{5}, \pm\frac{9}{5}\right)\)</p>
<p>(d) \(\left(\pm\frac{12}{5}, m\frac{9}{5}\right)\)</p>
Step-by-Step Solution
Key Concept: For maximum common chord length, the line of centers is perpendicular to the common chord; use this geometric constraint along with the radius condition.
<p>For maximum length of common chord, the line of centers must be perpendicular to the common chord. Since the common chord has slope \(\frac{3}{4}\), the line of centers has slope \(-\frac{4}{3}\). The common chord is a diameter of the smaller circle \(C_1\) when the length is maximum (both circles intersect with maximum overlap). Using the perpendicularity condition and the constraint that \(C_2\) has radius 5, the center of \(C_2\) lies on the line through origin with slope \(-\frac{4}{3}\) at distance satisfying the radical axis condition.</p>
Correct Answer: A