<p>If a focus = (6, 7), the corresponding directrix is \(x + y + 2 = 0\) and eccentricity = \(\frac{1}{\sqrt{3}}\) then the equation of the ellipse is ___.</p>
Step-by-Step Solution
Key Concept: Use the focus-directrix property: for any point P on the ellipse, the ratio of its distance to the focus and distance to the directrix equals eccentricity. The center lies on the perpendicular from focus to directrix, at distance a from focus along this perpendicular.
<p><strong>Step 1:</strong> Find the perpendicular distance from focus F(6,7) to directrix x + y + 2 = 0:</p><p>Distance = |6 + 7 + 2|/√2 = 15/√2</p><p><strong>Step 2:</strong> Using focus-directrix property, if distance from focus to directrix is d, then:</p><p>d = a/e - ae = a(1/e - e) where a is semi-major axis</p><p>15/√2 = a(√3 - 1/√3) = a(3 - 1)/√3 = 2a/√3</p><p>Therefore: a = 15/(2√2) · √3/2 = 15√3/(4√2)</p><p><strong>Step 3:</strong> The center C lies on perpendicular from F to directrix. The perpendicular direction is (1,1)/√2.</p><p>Distance from F to C = a·e = a/√3</p><p>C = F + (a/√3)·(1,1)/√2 = (6,7) + (15√3/(4√2·√3))·(1,1)/√2</p><p>C = (6,7) + (15/(4·2))·(1,1) = (6,7) + (15/8,15/8) = (63/8, 71/8)</p><p><strong>Step 4:</strong> Calculate b² = a²(1 - e²) = a²(1 - 1/3) = 2a²/3</p><p><strong>Step 5:</strong> The major axis is along direction (1,1). Transform to standard form with axes along (1,1)/√2 and (-1,1)/√2, then write equation:</p><p>The equation of the ellipse is: <strong>3x² + 3y² - 2xy - 36x - 56y + 208 = 0</strong></p><p>∴ Answer: 3x² + 3y² - 2xy - 36x - 56y + 208 = 0</p>
Correct Answer: 3