Question

If the foci of the ellipse \[ \frac{x^2}{25}+\frac{y^2}{k^2}=1 \qquad (k^2<25) \] and the hyperbola \[ \frac{x^2}{k}-\frac{y^2}{5}=1 \] are the same, then the product of the lengths of the latus rectum of the ellipse and that of the hyperbola is:

• A. \(25\)
• B. \(50\)
• C. \(16\)
• D. \(32\)

Hint:

Remember the latus rectum formulas: For an ellipse: \[ L=\frac{2b^2}{a} \] For a hyperbola: \[ L=\frac{2b^2}{a} \] The formula is the same; only the values of \(a\) and \(b\) differ according to the conic.

Answer 1

The Correct Option is D