If $P(\theta)=\left(x_1, \frac{3 \sqrt{5}}{2}\right), 0<\theta<\frac{\pi}{2}$ is a point on the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, where $\theta$ is the parameter in its parametric form, then $2 x_1+9 \sin ^2 \theta=$

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ( $k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

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