Let $P$ be any point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If $S_1$ and $S_2$ are its foci,then the maximum area of $\Delta PS_1S_2$ is (in square units):

The equation $\frac{x^{2}}{2-\lambda}-\frac{y^{2}}{\lambda-5}-1=0$ represents an ellipse,if

Let $A, A^{\prime}$ be the end points of the major axis,$S, S^{\prime}$ be the foci,and $B, B^{\prime}$ be the end points of the minor axis of an ellipse $E$. If $\angle BAB^{\prime}=60^{\circ}$,then find $\angle SBS^{\prime}$.

Find the area of the quadrilateral formed by the tangents at the endpoints of the latus rectum of the ellipse $x^2 + 2y^2 = 2$.

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} SB = \frac{\pi}{6}$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the ellipse $E$,then the sum of the squares of the lengths of major axis and minor axis is

The centre of the ellipse $4x^2 + 9y^2 - 16x - 54y + 61 = 0$ is