# The length of the major axis of an ellipse is 20 units, and its foci

Question:

The length of the major axis of an ellipse is 20 units, and its foci are $(\pm 5 \sqrt{3}, 0)$ Find the equation of the ellipse.

Solution:

Let the equation of the required ellipse be

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Given: Length of Major Axis = 20units ...(i)

We know that,

Length of Major Axis $=2 \mathrm{a}$...(ii)

∴ From eq. (i) and (ii), we get

$2 a=20$

$\Rightarrow a=10$

It is also given that,

Coordinates of foci $=(\pm 5 \sqrt{3}, 0)$...(iii)

We know that,

Coordinates of foci $=(\pm \mathrm{c}, 0)$...(iv)

$\therefore$ From eq. (iii) and (iv), we get

$c=5 \sqrt{3}$

We know that,

$c^{2}=a^{2}-b^{2}$

$\Rightarrow(5 \sqrt{3})^{2}=(10)^{2}-b^{2}$

$\Rightarrow 75=100-b^{2}$

$\Rightarrow b^{2}=100-75$

$\Rightarrow b^{2}=25$

Substituting the value of $a^{2}$ and $b^{2}$ in the equation of an ellipse, we get

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

$\Rightarrow \frac{x^{2}}{100}+\frac{y^{2}}{25}=1$