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

$S^{\prime}$ is the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{b^2}=1, (b < 5)$ lying on the negative $X$-axis and $P(\theta)$ is a point on this ellipse. If the distance between the foci of this ellipse is $8$ and $S^{\prime}P = 7$,then $\theta =$

The foci of $16x^2 + 25y^2 = 400$ are

Let $E_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a > b$. Let $E_{2}$ be another ellipse such that it touches the end points of the major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of the minor axis of $E_{1}$. If $E_{1}$ and $E_{2}$ have the same eccentricity $e$,then the value of $e$ is:

If the length of the minor axis of an ellipse is equal to half of the distance between the foci,then the eccentricity of the ellipse is:

The number of values of $c$ such that the line $y=4x+c$ touches the curve $\frac{x^{2}}{4}+y^{2}=1$ is