The equation of the ellipse whose one vertex is $(0, 7)$ and the directrix is $y = 12$ is:

Find the equation of the ellipse which passes through the points $(-3, 1)$ and $(2, -2)$,whose center lies at $(0, 0)$ and major axis lies along the $X$-axis.

The product of the lengths of the perpendiculars drawn from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ to the tangent at any point on the ellipse is

Consider two straight lines,each of which is tangent to both the circle $x^2 + y^2 = \frac{1}{2}$ and the parabola $y^2 = 4x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O(0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$,then which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ For the ellipse,the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$.
$(B)$ For the ellipse,the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$.
$(C)$ The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{4\sqrt{2}}(\pi - 2)$.
$(D)$ The area of the region bounded by the ellipse between the lines $x = \frac{1}{\sqrt{2}}$ and $x = 1$ is $\frac{1}{16}(\pi - 2)$.

Let $P$ be any point on the ellipse $7x^2 + 16y^2 = 112$,$S$ be a focus,$L$ be the corresponding directrix,and $PM$ be the perpendicular distance from $P$ to the directrix $L$. Then $\frac{SP}{PM} =$

If $A = \{(x, y) : x^2 + y^2 = 25\}$ and $B = \{(x, y) : x^2 + 9y^2 = 144\}$,then the number of points in $A \cap B$ is