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Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)

Question:

Find the equation for the ellipse that satisfies the given conditions: Vertices $(\pm 5,0)$, foci $(\pm 4,0)$,

Solution:

Vertices $(\pm 5,0)$, foci $(\pm 4,0)$

Here, the vertices are on the x-axis.

Therefore, the equation of the ellipse will be of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where $a$ is the semi-major axis.

Accordingly, $a=5$ and $c=4$.

It is known that $a^{2}=b^{2}+c^{2}$.

$\therefore 5^{2}=b^{2}+4^{2}$

$\Rightarrow 25=b^{2}+16$

$\Rightarrow b^{2}=25-16$

$\Rightarrow b=\sqrt{9}=3$

Thus, the equation of the ellipse is $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$ or $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$.

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