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