Question

If the centre, one of the foci and semi-major axis of an ellipse be $(0,0),(0,3)$and $5$ , then its equation is:

• A. $\frac{x^2}{16}+\frac{y^2}{25}=1$
• B. $\frac{x^2}{25}+\frac{y^2}{16}=1$
• C. $\frac{x^2}{9}+\frac{y^2}{25}=1$
• D. none of these

Answer 1

The correct option is A

$\frac{x^2}{16}+\frac{y^2}{25}=1$

Explanation for the correct option.

Given center $\left(0,0\right)$, focus $(0,3),$semi major axis i.e; $\mathrm{b}=5$.

The general equation of ellipse is :

$\frac{\left(x-h\right)}{a^2}+\frac{\left(y-k\right)}{b^2}=1$; where $\left(h,k\right)$ is the center & $a,b$are the length major & minor axis.

$$\begin{gathered} \Rightarrow \mathrm{be}=3 \\ \Rightarrow \mathrm{e}=\frac{3}{5} \end{gathered}$$

$\begin{array}{rcl}\mathrm{a}& =& \mathrm{b}\sqrt{1-{\mathrm{e}}^2}\\ \mathrm{a}& =& 5\sqrt{1-\frac{9}{25}}\\ & =& 4\end{array}$
So, the equation of ellipse is $\frac{x^2}{16}+\frac{y^2}{25}=1$.

Hence, option Ais correct.