Circles
Radical Axis
Grade 11
Question:
<p>Consider the circles \(C_1 \equiv x^2 + y^2 - 2x - 4y - 4 = 0\) and \(C_2 \equiv x^2 + y^2 + 2x + 4y + 4 = 0\) and the line \(L \equiv x + 2y + 2 = 0\), then</p>
<p>L is the radical axis of \(C_1\) and \(C_2\)</p>
<p>L is the common tangent of \(C_1\) and \(C_2\)</p>
<p>L is the common chord of \(C_1\) and \(C_2\)</p>
<p>L is perpendicular to the joining centers of \(C_1\) and \(C_2\)</p>
Step-by-Step Solution
Key Concept: Find the center and radius of each circle by converting to standard form, then determine the geometric relationship between the circles and line by checking distances from centers to the line relative to their radii.
<p><strong>Step 1: Convert C₁ to standard form</strong></p><p>C₁: x² + y² - 2x - 4y - 4 = 0</p><p>(x² - 2x + 1) + (y² - 4y + 4) - 1 - 4 - 4 = 0</p><p>(x - 1)² + (y - 2)² = 9</p><p>Center O₁ = (1, 2), Radius r₁ = 3</p><p><strong>Step 2: Convert C₂ to standard form</strong></p><p>C₂: x² + y² + 2x + 4y + 4 = 0</p><p>(x² + 2x + 1) + (y² + 4y + 4) + 4 - 1 - 4 = 0</p><p>(x + 1)² + (y + 2)² = 1</p><p>Center O₂ = (-1, -2), Radius r₂ = 1</p><p><strong>Step 3: Find distance from O₁ to line L</strong></p><p>d₁ = |1 + 2(2) + 2|/√(1² + 2²) = |1 + 4 + 2|/√5 = 7/√5 ≈ 3.13</p><p>Since d₁ > r₁, line L does NOT intersect C₁</p><p><strong>Step 4: Find distance from O₂ to line L</strong></p><p>d₂ = |-1 + 2(-2) + 2|/√5 = |-1 - 4 + 2|/√5 = 3/√5 ≈ 1.34</p><p>Since d₂ > r₂, line L does NOT intersect C₂</p><p><strong>Step 5: Verify circle relationship</strong></p><p>Distance between centers = √[(1-(-1))² + (2-(-2))²] = √(4 + 16) = 2√5 ≈ 4.47</p><p>Since r₁ + r₂ = 4 and 2√5 > 4, circles are external to each other</p><p>∴ Answer: A (The line does not intersect either circle, and the circles are external)</p>
Correct Answer: A