Interaction between Two Conics

A circle of radius $2$ units is drawn through the points in which the hyperbola $\frac{x^2}{3}-\frac{y^2}{2}=1$ and the ellipse $\frac{x^2}{a^2}+y^2=1$ intersect length of major axis of the ellipse is

Maximum number of common normals of $y^2=4ax$ and $x^2=4by$ is equal to

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P\left(x_1, y_2\right), Q\left(x_2, y_2\right), R\left(x_3, y_3\right), S\left(x_4, y_4\right),$ then

The two conics $b{x}^2=y$ and $\frac{x^2}{a^2}-\mathrm{ }\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\mathrm{ }1$ intersect if $a$ is 

If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ coincide, then $b^2$ is equal to

A circle has the same centre as an ellipse and passes through the foci $F_1 \& F_2$ of the ellipse, such that the two curves intersect in $4$ points. Let $P$ be any one of their point of intersection. If the major axis of the ellipse is $17$ & the area of the triangle $P{F}_1{F}_2$ is $30$, then the distance between the foci is -

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

The Minimum distance between the circles $x^2+y^{2 }=9$ and the curve $2x^2+10y^2+6xy=1$ is :

Given an ellipse whose equation is $\frac{{\text{x}}^2}{9}+\frac{{\text{y}}^2}{4}=1$. A tangent is drawn at any point P on the ellipse. This tangent at the point P intersects the tangents at the extremities of the major axes i.e., x = + 3 and x = -3 at T and T', respectively. Now, taking TT' as diameter, a circle is constructed. For every point P on the ellipse, this circle will always pass through which of the following points ?   

Equation of common tangent between circle $x^2+y^2=16$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{4}=1$ in the first quadrant is