Find the equation to the ellipse in the following case:Foci ( 3, … - Vidyadip

Question

Find the equation to the ellipse in the following case:Foci $(\pm3, 0), a = 4$

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Answer

Let the equation of the required ellipse be$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1\ \dots(\text{i})$
we have, Length of major axis = 16$\text{a}=4$
$\Rightarrow\text{a}^2=16$
and, the coordinates of foci are $(\pm3, 0)$
$\therefore\ \text{ae}=3$
$\Rightarrow4\times\text{e}=3$
$\Rightarrow\text{e}=\frac{3}{4}$
Now, $\text{b}^2=\text{a}^2(1-\text{e}^2)$
$=4^2\Big[1-\Big(\frac{3}{4}\Big)^2\Big]$
$=16\times\Big(1-\frac{9}{16}\Big)$
$=16\times\frac{7}{16}$
$=7$ Substituting the values of $a^2$ and $d^2$ in (i), we get$\frac{\text{x}^2}{16}+\frac{\text{y}^2}{7}=1$
This is the equation of the required ellipse.

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