If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}}\, + \,\frac{{{y^2}}}{{{b^2}}}\, = \,1(a\, > \,b)$  is twice the area of the ellipse, then the eccentricity of the  ellipse is

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Tangents are drawn from points onthe circle $x^2 + y^2 = 49$ to the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{24}} = 1$ angle between the tangents is

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2$ is $.......$

The foci of the ellipse $25{(x + 1)^2} + 9{(y + 2)^2} = 225$ are at

If $m$ is the slope of a common tangent to the curves $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and $x^{2}+y^{2}=12$, then $12\; m ^{2}$ is equal to