$B$ is an extremity of the minor axis of an ellipse whose foci are $S$ and $S^{\prime}$. If $\angle SBS^{\prime}$ is a right angle, then the eccentricity of the ellipse is

The area of the triangle formed by the tangent and normal to the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $P\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$ with the $x$-axis is

Let $S$ and $S^{\prime}$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(SP)^2+(S^{\prime}P)^2-SP \cdot S^{\prime}P=37$, then $\alpha^2+\beta^2$ is equal to:

The locus of the midpoints of the portion of the tangents of the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ intercepted between the coordinate axes is

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then the average of $\cos(\theta_1-\theta_2)$,$\cos(\theta_2-\theta_3)$ and $\cos(\theta_3-\theta_1)$ is

The eccentricity of the ellipse $(x - 3)^2 + (y - 4)^2 = \frac{y^2}{9}$ is