Find the equation of the ellipse whose foci are at \[\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)\] and \[e=\frac{4}{5}\]

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Find the equation of the ellipse whose foci are at

$\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)$

and

$e=\frac{4}{5}$

CBSE | Class 11 | Ellipse | Maths | RS Aggarwal

Find the equation of the ellipse whose foci are at

$\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)$

and

$e=\frac{4}{5}$

Given:

Coordinates of foci =

$\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)$

…(i)

We know that,

Coordinates of foci =

$\left( 0,\text{ }\pm c \right)$

…(ii)

The coordinates of the foci are

$\left( \mathbf{0},\text{ }\pm \mathbf{4} \right)$

.

This means that the major and minor axes are along y and x axes respectively.

∴ From eq. (i) and (ii), we get

c =

$4$

It is also given that

Eccentricity =

$\frac{4}{5}$

we know that,

Eccentricity e = c/a

$\frac{4}{5}=\frac{4}{a}\,[c=4]$

⇒ a =

$5$

Now, we know that,

$\begin{array}{*{35}{l}} {{c}^{2}}=\text{ }{{a}^{2}}\text{ }{{b}^{2}} \\ \Rightarrow {{\left( 4 \right)}^{2}}=\text{ }{{\left( 5 \right)}^{2}}\text{ }{{b}^{2}} \\ \Rightarrow 16\text{ }=\text{ }25\text{ }\text{ }{{b}^{2}} \\ \Rightarrow {{b}^{2}}=\text{ }25\text{ }\text{ }16 \\ \Rightarrow {{b}^{2}}=\text{ }9 \\ \end{array}$