Question

The locus of mid points of tangents intercepted between the axes of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is

• A. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \)
• B. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
• C. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 4 \)
• D. \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 3 \)

Hint:

The locus of midpoints of tangents to an ellipse is another ellipse with a scaled equation.

Answer 1

The Correct Option is A

Step 1: Understanding the locus.
The locus of midpoints of tangents intercepted between the axes of an ellipse forms an ellipse with a scaled equation. This scaled ellipse has a factor of 2 in the equation.

Step 2: Conclusion.
The correct equation for the locus is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \), corresponding to option (1).