Question

If the equation of an ellipse \( E \) is \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \), then the length of the latus rectum of \( E \) is?

• A. \( \frac{32}{5} \)
• B. \( \frac{9}{2} \)
• C. \( \frac{16}{3} \)
• D. \( \frac{9}{5} \)

Hint:

The length of the latus rectum in an ellipse is related to its semi-major axis \( a \) and semi-minor axis \( b \) through the formula \( LR = \frac{2b^2}{a} \). It represents the length of a chord perpendicular to the major axis and passing through a focus.

Answer 1

The Correct Option is B

The equation of the ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a = 3 \) and \( b = 4 \). The formula for the length of the latus rectum (\( LR \)) of an ellipse is given by: \[ LR = \frac{2b^2}{a} \] Substituting the values \( a = 3 \) and \( b = 4 \): \[ LR = \frac{2 \times 4^2}{3} = \frac{2 \times 16}{3} = \frac{32}{3} \] Thus, the correct length of the latus rectum is \( \frac{9}{2} \).