Question

If \( CP \) and \( CD \) is a pair of semi-conjugate diameters of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), then \( CP^2 + CD^2 = \)

• A. \( \frac{a^2 + b^2}{2} \)
• B. \( a^2 + b^2 \)
• C. \( a^2 - b^2 \)
• D. \( \frac{a^2 - b^2}{2} \)

Hint:

In ellipse geometry, conjugate diameters are related by the equation \( CP^2 + CD^2 = a^2 + b^2 \).

Answer 1

The Correct Option is B

Step 1: Use the property of conjugate diameters.
For an ellipse with semi-conjugate diameters \( CP \) and \( CD \), the relation between the diameters is given by: \[ CP^2 + CD^2 = a^2 + b^2 \]
Step 2: Conclusion.
The correct answer is \( a^2 + b^2 \).