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 $2 a$ and $2 b$ 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 $C A=a$
$\Rightarrow \sqrt{(4-2)^{2}+(-3+3)^{2}}=a$
$\Rightarrow a=2$ ...(ii)
Here, $C S=a e$
$\Rightarrow \sqrt{(2-3)^{2}+(-3+3)^{2}}=a e$
$\Rightarrow a e=1$ ...(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
$\frac{(x-2)^{2}}{4}+\frac{(y+3)^{2}}{3}=1$