If the tangents drawn from a point $P$ to the ellipse $4x^2+9y^2-16x+54y+61=0$ are perpendicular,then the locus of $P$ is

The equation of an ellipse whose focus is $(-1, 1)$,whose directrix is $x - y + 3 = 0$,and whose eccentricity is $e = \frac{1}{2}$,is given by

The eccentricity of the conic $4x^2 + 16y^2 - 24x - 3y = 1$ is

Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let $S(p, q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two tangents are drawn from $S$ to the ellipse,of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $T$ in the fourth quadrant. Let $R$ be the vertex of the ellipse with positive $x$-coordinate and $O$ be the center of the ellipse. If the area of the triangle $\triangle ORT$ is $\frac{3}{2}$,then which of the following options is correct?

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

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