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10-2. Parabola, Ellipse, Hyperbola
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Find the equation for the ellipse that satisfies the given conditions : Vertices $(\pm 5,\,0),$ foci $(\pm 4,\,0)$
Option A
Option B
Option C
Option D
Solution
Vertices $(\pm 5,\,0),$ foci $(±4,\,0)$
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 $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$
Standard 11
Mathematics
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