Question:
The eccentricity of an ellipse is \( \frac{1}{3} \) and its center is at the origin. If one of the directrices is \( x = 9 \), then the equation of the ellipse is:
The eccentricity of an ellipse is \( \frac{1}{3} \) and its center is at the origin. If one of the directrices is \( x = 9 \), then the equation of the ellipse is:
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For ellipses, use the relationship between the eccentricity and the directrix to solve for the equation.
Updated On: Mar 7, 2025
- \( 8x^2 + 9y^2 = 32 \)
- \( 8x^2 + 9y^2 = 36 \)
- \( 9x^2 + 8y^2 = 36 \)
- \( 9x^2 + 8y^2 = 32 \)
- \( 8x^2 + 9y^2 = 72 \)
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Solution and Explanation
For an ellipse with eccentricity \( e = \frac{1}{3} \) and the equation of the directrix \( x = 9 \), the equation of the ellipse is given by:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Substitute the values and simplify to get the equation \( 8x^2 + 9y^2 = 72 \).
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