<p>The angle between a pair of tangents drawn from a point \(P\) to the circle \(x^2 + y^2 + 4x - 6y + 9\sin^2\alpha + 13\cos^2\alpha = 0\) is \(2\alpha\). The equation of the locus of the point \(P\) is</p>
<p>\(x^2 + y^2 + 4x - 6y + 4 = 0\)</p>
<p>\(x^2 + y^2 + 4x - 6y - 9 = 0\)</p>
<p>\(x^2 + y^2 + 4x - 6y - 4 = 0\)</p>
<p>\(x^2 + y^2 + 4x - 6y + 9 = 0\)</p>
Step-by-Step Solution
Key Concept: The angle between two tangents from external point P to a circle relates to the distance of P from center via: if angle is 2α, then sin(α) = r/d, where r is radius and d is distance from center. Use this relationship combined with the parametric circle equation to find the locus.
<p><strong>Step 1: Rewrite the circle equation in standard form</strong></p><p>x² + y² + 4x - 6y + 9sin²α + 13cos²α = 0</p><p>(x+2)² + (y-3)² - 4 - 9 + 9sin²α + 13cos²α = 0</p><p>(x+2)² + (y-3)² = 13 - 9sin²α - 13cos²α = 13 - 9sin²α - 13(1-sin²α)</p><p>(x+2)² + (y-3)² = 13 - 9sin²α - 13 + 13sin²α = 4sin²α</p><p>Center C = (-2, 3), Radius r = 2|sin α|</p><p><strong>Step 2: Apply tangent angle formula</strong></p><p>If angle between tangents from P is 2α, and if ∠TPC = α (where T is point of tangency), then:</p><p>sin(α) = r/CP, where CP is distance from P to center C</p><p>sin(α) = 2sin(α)/CP</p><p>CP = 2</p><p><strong>Step 3: Write the locus equation</strong></p><p>Points P at constant distance 2 from C(-2, 3) form a circle:</p><p>(x+2)² + (y-3)² = 4</p><p>∴ Answer: A</p>
Correct Answer: A