Question

If the semi-major axis of an ellipse is \( 3 \) and the latus rectum is \( \frac{16}{9} \), then the standard equation of the ellipse is

• A. \( \frac{x^2}{9} + \frac{y^2}{2} = 1 \)
• B. \( \frac{x^2}{8} + \frac{y^2}{9} = 1 \)
• C. \( \frac{x^2}{9} + \frac{3y^2}{8} = 1 \)
• D. \( \frac{3x^2}{8} + \frac{y^2}{9} = 1 \)
• E. \( \frac{x^2}{9} + \frac{8y^2}{3} = 1 \)

Hint:

Always remember latus rectum formula \( \frac{2b^2}{a} \) for ellipse.

Answer 1

The Correct Option is C

Concept:
• Equation of ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
• Length of latus rectum: \[ \frac{2b^2}{a} \]

Step 1: Identify given values.
\[ a = 3 \Rightarrow a^2 = 9 \] \[ \text{Latus rectum} = \frac{16}{9} \]

Step 2: Use formula.
\[ \frac{2b^2}{a} = \frac{16}{9} \] \[ \frac{2b^2}{3} = \frac{16}{9} \]

Step 3: Solve for \(b^2\).
\[ 2b^2 = \frac{16}{3} \Rightarrow b^2 = \frac{8}{3} \]

Step 4: Write equation.
\[ \frac{x^2}{9} + \frac{y^2}{8/3} = 1 \] Multiply numerator: \[ \frac{x^2}{9} + \frac{3y^2}{8} = 1 \]

Step 5: Final Answer.
\[ \boxed{(C)} \]