Equation of the ellipse whose minor axis is equal to the distance between foci and whose latus rectum is $10 ,$ is given by ____________.

The standard equation of ellipse is,

$\Rightarrow \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text {. }$

According to the question, the minor axis is equal to the distance between the foci.

$\begin{array}{l} \Rightarrow 2 b=2 a e \text { and } \\ \Rightarrow \frac{2 b^{2}}{a}=10 \\ \Rightarrow b^{2}=5 a \\ \Rightarrow b=a e \\ \Rightarrow b^{2}=a^{2} e^{2} \\ \Rightarrow b^{2}=a^{2}\left(1-\frac{b^{2}}{a^{2}}\right) \quad\left[\because e=\sqrt{1-\frac{b^{2}}{a^{2}}}\right] \\ \Rightarrow b^{2}=a^{2}-b^{2} \\ \Rightarrow a^{2}=2 b^{2} \\ \left.\Rightarrow a^{2}=10 a \quad \quad \quad \quad (\because b^{2}=5 a\right) \\ \Rightarrow a=10 \\ \Rightarrow b^{2}=5 a \end{array}$

$\therefore b^{2}=50$

Substituting the values of ' $a$ ' and ' $b$ ' in the equation of an ellipse, we get:

$\begin{array}{l} \Rightarrow \frac{x^{2}}{100}+\frac{y^{2}}{50}=1 \\ \therefore x^{2}+2 y^{2}=100 . \end{array}$

This is the required equation of the ellipse.