Question

The equation of the ellipse whose axes are parallel to the coordinate axes having its centre at the point $(2,-3)$ and focus at $(3,-3)$ and one vertex at $(4,-3)$ is

• A. $\frac{(x-2{)}^2}{4}+\frac{(y+3{)}^2}{3}=1$
• B. $\frac{(x+2{)}^2}{3}+\frac{(y+3{)}^2}{4}=1$
• C. $\frac{(x+2{)}^2}{4}+\frac{(y+3{)}^2}{3}=1$
• D. None of the above

Answer 1

Answer: (A)

Let $2a$ and $2b$ be the major and minor axes of the ellipse. Then, its equation is
$\frac{(x-2{)}^2}{a^2}+\frac{(y+3{)}^2}{b^2}=1$ ...(i)

Here, semi-major axis $CA=a$
$\Rightarrow \sqrt{(4-2{)}^2+(-3+3{)}^2}=a$
$\Rightarrow a=2$ ...(ii)
Here, $CS=ae$
$\Rightarrow \sqrt{(2-3{)}^2+(-3+3{)}^2}=ae$
$\Rightarrow ae=1$ ...(iii)
From Eqs. (ii) and (iii), we get
$e=\frac{1}{2}$
Now, $b^2=a^2\left(1-e^2\right)$
$\Rightarrow b^2=4\left(1-\frac{1}{4}\right)=3$
On substituting the values of $a$ and $b$ in Eq. (i) we get
$\frac{(x-2{)}^2}{4}+\frac{(y+3{)}^2}{3}=1$