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 $S{S}^{\prime }=2ae\Rightarrow {\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2=4(2{)}^2{\left(\frac{\sqrt{3}}{2}\right)}^2$
$\Rightarrow {\left(x_1+x_2\right)}^2+{\left(y_1+y_2\right)}^2-4\left(x_1{x}_2+y_1{y}_2\right)=12\Rightarrow (2h{)}^2+(2k{)}^2-4(1+1)=12$
$($ 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 $x^2+y^2=5$