Circles
Concentric Circles
Grade 11
Question:
<p>If <strong>(–3, 2)</strong> lies on the circle <strong>x² + y² + 2gx + 2fy + c = 0</strong> which is concentric with the circle <strong>x² + y² + 6x + 8y - 5 = 0</strong>, then <strong>c</strong> is equal to</p>
<p>(a) 11</p>
<p>(b) –11</p>
<p>(c) 24</p>
<p>(d) 100</p>
Step-by-Step Solution
Key Concept: Concentric circles differ only in their constant term. Substitute the given point into the general form to find the constant.
<p><strong>Step 1:</strong> The equation of the family of concentric circles to the circle <strong>x² + y² + 6x + 8y - 5 = 0</strong> is:</p><p><strong>x² + y² + 6x + 8y + λ = 0</strong></p><p><strong>Step 2:</strong> Since the point <strong>(–3, 2)</strong> lies on this circle, substitute the coordinates:</p><p><strong>(–3)² + (2)² + 6(–3) + 8(2) + c = 0</strong></p><p><strong>Step 3:</strong> Simplify:</p><p><strong>9 + 4 - 18 + 16 + c = 0</strong></p><p><strong>11 + c = 0</strong></p><p><strong>∴ c = –11</strong></p><p>Answer is <strong>(b)</strong>.</p>
Correct Answer: B