Ellipse
Director Circle
Grade 11
Question:
<p>Find the equation of the director circle of the ellipse <span>\(\frac{x^2}{5} + \frac{y^2}{3} = 1\)</span></p>
<p>(a) \(x^2 + y^2 = 2\)</p>
<p>(b) \(x^2 + y^2 = 5\)</p>
<p>(c) \(x^2 + y^2 = 8\)</p>
<p>(d) \(x^2 + y^2 = 15\)</p>
Step-by-Step Solution
Key Concept: The director circle of an ellipse has radius equal to the distance from center to any point where tangents are perpendicular. For ellipse with semi-major axis a and semi-minor axis b, the director circle equation is x² + y² = a² + b².
<p><strong>Solution:</strong> For an ellipse <span>\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)</span>, the director circle is <span>\(x^2 + y^2 = a^2 + b^2\)</span>.</p><p>Here, <span>\(a^2 = 5\)</span> and <span>\(b^2 = 3\)</span></p><p>Therefore, the equation of director circle is <span>\(x^2 + y^2 = 5 + 3 = 8\)</span></p><p>∴ Answer is (c)</p>
Correct Answer: c