Let $A\left(\theta_1\right)$ and $B\left(\theta_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola, If $A, S, B$ are collinear and

a $\cos \left(\frac{\theta_1+\theta_2}{2}\right)=k \cos \left(\frac{\theta_1-\theta_2}{2}\right)$, then $k=$

If $p, q$ are the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola respectively, then the area of the square (in sq. units) formed by the points of intersection of the ellipse $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ and the pair of lines $x^2-y^2=0$ is

Suppose the axes $X$ and $Y$ are obtained by rotating the axes $x$ and $y$ an angle $\theta$. If the equation $x^2+2 \sqrt{3} x y-y^2=4 a^2$ is transformed to $X^2-Y^2=2 a^2$ with respect to the $X Y$-axes, then $\theta$ is equal to

For the hyperbola $x^2-y^2-4 x+2 y+c=0$, if the focus is $S(2+2 \sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$, then $c=$