Question

The locus of the extremities of the latus rectum of the family of ellipses \(b^2x^2 + y^2 = a^2b^2\) having a given major axis is

• A. \(x^2 + ay = a^2\)
• B. \(y^2 + bx = a^2\)
• C. \(x^2 \pm ay = a^2\)
• D. None of these

Hint:

Endpoints of latus rectum for ellipse: \((\pm a, \frac{b^2}{a})\). Substitute \(b^2\) to get locus.

Answer 1

The Correct Option is A

Concept: Given ellipse: \[ b^2x^2 + y^2 = a^2b^2 \Rightarrow \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Major axis is fixed $\Rightarrow$ \(a\) is constant.

Step 1: Coordinates of latus rectum endpoints.
For ellipse: \[ \text{Endpoints of latus rectum} = (\pm a, \frac{b^2}{a}) \]

Step 2: Eliminate parameter \(b\).
\[ y = \frac{b^2}{a} \Rightarrow b^2 = ay \]

Step 3: Substitute into ellipse.
\[ \frac{x^2}{a^2} + \frac{y^2}{ay} = 1 \] \[ \frac{x^2}{a^2} + \frac{y}{a} = 1 \] Multiply by \(a^2\): \[ x^2 + ay = a^2 \] Conclusion: \[ {x^2 + ay = a^2} \]