An arch is in the form of a semi-ellipse. It is $8 \, m$ wide and $2 \, m$ high at the centre. Find the height of the arch at a point $1.5 \, m$ from one end. (in $, m$)

The equations $x = a \cos \theta$ and $y = b \sin \theta$ with $a > b$ represent a conic section. Its eccentricity $e$ is given by:

$A$ tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ intersects the coordinate axes at $A$ and $B$. Then the locus of the circumcentre of triangle $AOB$ (where $O$ is the origin) is:

The distance between the directrices of the ellipse $\frac{x^2}{36}+\frac{y^2}{20}=1$ is

The eccentricity of the ellipse $4x^2 + 25y^2 = 100$ is

Let $S(1,0)$ and $S^{\prime}(0,1)$ be the foci of an ellipse such that $SP+S^{\prime} P=2$ for any point $P$ on the ellipse. If $A(x_1, y_1)$ and $A^{\prime}(x_2, y_2)$ are the end points of the major axis of this ellipse,then $x_1+x_2=$