Question

The equations of the directrices of the ellipse \(9x^2 + 4y^2 - 18x - 16y - 11 = 0\) are:

• A. \(y = 2 \pm \frac{9}{\sqrt{5}}\)
• B. \(x = 1 \pm \frac{6}{\sqrt{5}}\)
• C. \(x = 2 \pm \frac{9}{\sqrt{5}}\)
• D. \(y = 1 \pm \frac{6}{\sqrt{5}}\)

Hint:

For an ellipse, the directrices depend on the orientation of the major axis and the eccentricity.
- For an ellipse with a vertical major axis, the directrices are of the form \(y = k \pm \frac{a}{e}\).

Answer 1

The Correct Option is A

To find the equations of the directrices of the ellipse, we first write the ellipse equation in its standard form by completing the square. The directrices of an ellipse with a vertical major axis are given by the equation \(y = k \pm \frac{a}{e}\), where \(a\) is the semi-major axis and \(e\) is the eccentricity. Using the properties of the ellipse, we find that the directrices are \(y = 2 \pm \frac{9}{\sqrt{5}}\). Thus, the correct answer is \(\boxed{y = 2 \pm \frac{9}{\sqrt{5}}}\).