Question

Directions: The following question has four choices, out of which one or more are correct.The equations of two ellipses are $\frac{x^2}{p^2}+y^2=1$ and $x^2+\frac{y^2}{p^2}=1,=1,$ where $p$ is a parameter. The locus of the points of intersection of both the ellipses is a set of curves comprising

• (A) two straight lines
• (B) one straight line
• (C) one circle
• (D) one parabola

Answer 1

The Correct Option is A, D

Explanation:
Let the point of intersections is $(h,k).$ Therefore,$$\begin{array}{l}\frac{h^2}{p^2}+k^2=1\text{ and }{h}^2+\frac{k^2}{p^2}=1\\ \frac{h^2}{p^2}=1-k^2\text{ and }1-h^2=\frac{k^2}{p^2}\\ \frac{h^2}{1-k^2}=p^2\text{ and }{p}^2=\frac{k^2}{1-h^2}\\ \frac{h^2}{1-k^2}=\frac{k^2}{1-h^2}\end{array}$$$$\begin{array}{rl}{h}^2(1-h^2)& =k^2(1-k^2)\\ h^2-h^4& =k^2-k^4\\ h^2-k^2-(h^4-k^4)& =0\\ (h^2-k^2)-(h^2-k^2)(h^2+k^2)& =0\\ (h^2-k^2)[1-(h^2+k^2)]& =0\\ h^2=k^2\mathrm{Or}{h}^2+k^2& =1\\ x^2=y^2\mathrm{Or}{x}^2+y^2& =1\\ x=\pm y\mathrm{Or}{x}^2+y^2& =1\end{array}$$Hence, it is the required solution.$x=\pm y$ represents two straight lines$x^2+y^2=1$ represents a circle