The centre of the ellipse $4{x^2} + 9{y^2} - 16x - 54y + 61 = 0$ is

In a triangle $A B C$ with fixed base $B C$, the vertex $A$ moves such that $\cos B+\cos C=4 \sin ^2 \frac{A}{2} .$ If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$, respectively, then

$(A)$ $b+c=4 a$

$(B)$ $b+c=2 a$

$(C)$ locus of point $A$ is an ellipse

$(D)$ locus of point $A$ is a pair of straight lines

The equations of the directrices of the ellipse $16{x^2} + 25{y^2} = 400$ are

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is

If two tangents drawn from a point $(\alpha, \beta)$ lying on the ellipse $25 x^{2}+4 y^{2}=1$ to the parabola $y^{2}=4 x$ are such that the slope of one tangent is four times the other, then the value of $(10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}$ equals

The distance between the foci of an ellipse is 16 and eccentricity is $\frac{1}{2}$. Length of the major axis of the ellipse is