The foci of $16{x^2} + 25{y^2} = 400$ are

The locus of the mid point of the line segment joining the point $(4,3)$ and the points on the ellipse $x^{2}+2 y^{2}=4$ is an ellipse with eccentricity

The  the circle passing through the foci of the $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$ and having centre at $(0,3) $ is

A tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{{{x^2}}}{{18}}$ + $\frac{{{y^2}}}{{32}}$ $= 1$  intersects the major and minor axes in points $A$ and $ B$  respectively. If $C$  is the centre of the ellipse then the area of the triangle $ ABC$  is : .............. $\mathrm{sq. \,units}$

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 $.......$

If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1, 2)$ and $(a, b)$ are perpendicular to each other, then $a^2$ is equal to