If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity is

Let $A = \left\{ {\left( {x,y} \right):\,y = mx + 1} \right\}$ 

      $B = \left\{ {\left( {x,y} \right):\,\,{x^2} + 4{y^2} = 1} \right\}$ 

$C = \left\{ {\left( {\alpha ,\beta } \right):\,\left( {\alpha ,\beta } \right) \in A\,\,and\,\,\left( {\alpha ,\beta } \right) \in B\,\,and\,\alpha \, > 0} \right\}$ . 

If set $C$ is singleton set then sum of all possible values of $m$ is

If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$ , then locus of $P(h, k)$ is a conic $C$ whose eccentricity equals

An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the probability that this point lies outside the ellipse is $\frac {2}{3}$ then eccentricity of ellipse is $\frac{{a\sqrt b }}{c}$ . Where $gcd( a, c) = 1$ and $b$ is square free integer ($b$ is not divisible by square of any integer except $1$ ) then $a · b · c$ is

If $a$ and $c$ are positive real numbers and the ellipse $\frac{{{x^2}}}{{4{c^2}}} + \frac{{{y^2}}}{{{c^2}}} = 1$ has four distinct points in common with the circle $x^2 + y^2 = 9a^2$ , then

If the line $x\cos \alpha + y\sin \alpha = p$ be normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, then