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 centre of ellipse is

• A. $x^2+y^2=3$
• B. $x ^2+ y ^2=5$
• C. $(x-2)^2+(y-1)^2=5$
• D. $( x -2)^2+( y -1)^2=3$

Answer 1

Answer: (B)

We know that $SS ^{\prime}=2 ae \Rightarrow\left( x _1- x _2\right)^2+\left( y _1- y _2\right)^2=4(2)^2\left(\frac{\sqrt{3}}{2}\right)^2$
$\Rightarrow\left( x _1+ x _2\right)^2+\left( y _1+ y _2\right)^2-4\left( x _1 x _2+ y _1 y _2\right)=12 \Rightarrow(2 h )^2+(2 k )^2-4(1+1)=12$
$\left(\right.$ As, $x_1 \cdot x_2$ and $y_1 y_2$ are perpendicular distances of the foci from their tangents $=b^2=1^2=1$ )
$\Rightarrow h ^2+ k ^2=5$
$\therefore$ The locus of centre $c ( h , k )$ of ellipse is $x ^2+ y ^2=5$