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?
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?
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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.
Updated On: Apr 4, 2025
- \( \frac{32}{5} \)
- \( \frac{9}{2} \)
- \( \frac{16}{3} \)
- \( \frac{9}{5} \)
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The Correct Option is B
Solution and Explanation
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 \( \frac{9}{2} \).
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