Let $E_1$ and $E_2$ be two ellipses $\frac{x^2}{a^2} + y^2 = 1$ and $x^2 + \frac{y^2}{a^2} = 1$ (where $a$ is a parameter). Then the locus of the points of intersection of the ellipses $E_1$ and $E_2$ is a set of curves comprising
Step-by-Step Solution
Key Concept: Finding the locus of intersection points requires eliminating the parameter $a^2$ from both ellipse equations. Substituting the two equations and simplifying algebraically yields the product form $(h-k)(h+k)(h^2+k^2-1)=0$, which decomposes into two straight lines ($y=x$ and $y=-x$) and one circle ($x^2+y^2=1$).
Given two ellipses $E_1: \frac{h^2}{a^2} + k^2 = 1$ and $E_2: \frac{h^2}{1} + \frac{k^2}{a^2} = 1$, eliminating $a^2$ from both equations yields $h^2(1-k^2) = k^2(1-k^2)$, which simplifies to $(h-k)(h+k)(h^2+k^2-1)=0$. Therefore, the locus is the union of three curves: the straight lines $y = x$, $y = -x$, and the circle $x^2 + y^2 = 1$.
Correct Answer: 1,3