Question

The length of the latus rectum of \( x^2 + 3y^2 = 12 \) is:

• A. \( \frac{1}{3} \) units
• B. \( \frac{4}{\sqrt{3}} \) units
• C. 24 units
• D. \( \frac{2}{3} \) units

Hint:

For ellipses, the length of the latus rectum can be found using \( \frac{2b^2}{a} \).

Answer 1

The Correct Option is B

The given equation is \( \frac{x^2}{12} + \frac{y^2}{4} = 1 \), which represents an ellipse. For an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the length of the latus rectum is given by: \[ LLR = \frac{2b^2}{a} \] From the given equation, \( a^2 = 12 \) and \( b^2 = 4 \), so: \[ LLR = \frac{2 \times 4}{\sqrt{12}} = \frac{8}{\sqrt{12}} = \frac{4}{\sqrt{3}} \] Thus, the length of the latus rectum is \( \frac{4}{\sqrt{3}} \) units.