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:

• A. \( 8x^2 + 9y^2 = 32 \)
• B. \( 8x^2 + 9y^2 = 36 \)
• C. \( 9x^2 + 8y^2 = 36 \)
• D. \( 9x^2 + 8y^2 = 32 \)
• E. \( 8x^2 + 9y^2 = 72 \)

Hint:

For ellipses, use the relationship between the eccentricity and the directrix to solve for the equation.

Answer 1

Answer: (E)

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 \( 8x^2 + 9y^2 = 72 \).