<p>For what values of <em>l</em> and <em>m</em> the circle \(5(x^2 + y^2) + ly - m = 0\) belongs to the coaxal system determined by the circles \(x^2 + y^2 + 2x + 4y - 6 = 0\) and \(2(x^2 + y^2) - x = 0\)?</p>
Step-by-Step Solution
Key Concept: A circle belongs to a coaxal system if it can be expressed as a linear combination of the two given circles: C₁ + λC₂ = 0. Expand and compare coefficients of x², y², x, y, and the constant term to find l and m.
<p><strong>Step 1:</strong> Write the two given circles in standard form by dividing to make coefficient of x² and y² unity:</p><p>C₁: x² + y² + 2x + 4y - 6 = 0</p><p>C₂: x² + y² - ½x = 0</p><p><strong>Step 2:</strong> Any circle in the coaxal system is: C₁ + λ(C₁ - C₂) = 0 or equivalently C₁ + λL = 0, where L is the radical axis.</p><p>Alternatively, use: C₁ + μC₂ = 0 where we combine them directly.</p><p><strong>Step 3:</strong> The radical axis of C₁ and C₂ is: (2x + 4y - 6) - (-½x) = 0 → 5x/2 + 4y - 6 = 0 or 5x + 8y - 12 = 0</p><p><strong>Step 4:</strong> Any member of coaxal system: x² + y² + 2x + 4y - 6 + λ(5x + 8y - 12) = 0</p><p>Rewrite given circle: 5(x² + y²) + ly - m = 0 as x² + y² + (l/5)y - m/5 = 0</p><p><strong>Step 5:</strong> Comparing x² + y² + 2x + 4y - 6 + λ(5x + 8y - 12) = 0 with x² + y² + (l/5)y - m/5 = 0:</p><p>Coefficient of x: 2 + 5λ = 0 → λ = -2/5</p><p>Coefficient of y: 4 + 8λ = l/5 → 4 + 8(-2/5) = l/5 → 4 - 16/5 = l/5 → 4/5 = l/5 → l = 4</p><p>Constant term: -6 - 12λ = -m/5 → -6 - 12(-2/5) = -m/5 → -6 + 24/5 = -m/5 → -6/5 = -m/5 → m = 6</p><p>∴ <strong>Answer: l = 4, m = 6</strong></p>
Correct Answer: l=4, m=6