Question

An ellipse has semi-major axis of length 2 and semi-minor axis of length 1 . It slides between the co-ordinate axes in the first quadrant while maintaining contact with both $x$-axis and $y$-axis.
The locus of the foci of the ellipse is

• A. $x^2+y^2+\frac{1}{x^2}+\frac{1}{y^2}=16$
• B. $x^2+y^2+\frac{1}{x^2}-\frac{1}{y^2}=2\sqrt{3}+4$
• C. $x^2+y^2-\frac{1}{x^2}-\frac{1}{y^2}=16$
• D. $x^2+y^2-\frac{1}{x^2}+\frac{1}{y^2}=2\sqrt{3}+4$

Answer 1

Answer: (A)

Now, $(2x-h)h=1\Rightarrow x=\left(\frac{1+h^2}{2h}\right);(2y-k)k=1\Rightarrow y=\left(\frac{1+k^2}{2k}\right)$
Also, $x^2+y^2=5\Rightarrow \frac{{\left(1+h^2\right)}^2}{4h^2}+\frac{{\left(1+k^2\right)}^2}{4k^2}=5$
$\therefore$ The locus of focus $S(h,k)$ is
$x^2+y^2+\frac{1}{x^2}+\frac{1}{y^2}=16$