Question

The equation of the ellipse centred at $(1,2)$, one focus at $(6,2)$ and passing through the point $(4,6),$ is

• A. $\frac{(x-1{)}^2}{45}+\frac{(y-2{)}^2}{20}=1$
• B. $\frac{(x-1{)}^2}{5}+\frac{(y-2{)}^2}{20}=1$
• C. $\frac{(x-1{)}^2}{20}+\frac{(y-2{)}^2}{45}=1$
• D. $\frac{(x-2{)}^2}{20}+\frac{(y-1{)}^2}{45}=1$

Answer 1

The correct option is A $\frac{(x-1{)}^2}{45}+\frac{(y-2{)}^2}{20}=1$


Let $S\equiv (6,2)$ and $C\equiv (1,2)$ , slope of $CS=0$, therefore major axis of the ellipse is parallel to $x-$axis and minor axis is parallel to $y-$axis.
As $C$ is the mid point of both foci
$\therefore$ other focus $S_1\equiv (-4,2)$
We know that $PS+P{S}_1=2a$
$2\sqrt{5}+4\sqrt{5}=2a\Rightarrow a^2=45$

Now $S{S}_1=2ae=10$
$a^2{e}^2=25=a^2-b^2\Rightarrow b^2=20$

Hence equation of required ellipse is
$\frac{(x-1{)}^2}{45}+\frac{(y-2{)}^2}{20}=1$