Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$ foci $(±5,\,0)$
Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$ foci $(±5,\,0)$
Length of major axis $=26 ;$ foci $=(\pm 5,\,0)$
since the foci are on the $x-$ axis, the major axis is along 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 semimajor axis.
Accordingly, $2 a=26 \Rightarrow a=13$ and $c=5$
It is known that $a^{2}=b^{2}+c^{2}$
$\therefore 13^{2}=b^{2}+5^{2}$
$\Rightarrow 169=b^{2}+25$
$\Rightarrow b^{2}=169-25$
$\Rightarrow b=\sqrt{144}=12$
Thus, the equation of the ellipse is $\frac{x^{2}}{13^{2}}+\frac{y^{2}}{12^{2}}=1$ or $\frac{x^{2}}{169}+\frac{y^{2}}{144}=1$
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