Let $O(0, 0)$ and $A(0, 1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $\Delta AOP$ is $4$ is:

Tangents are drawn from point $(1, 1)$ to the ellipse $S \equiv x^2 + 4y^2 - 2x + 8y + 1 = 0$. If $m_1, m_2$ $(m_1 > m_2)$ are the slopes of these tangents,then with respect to the given ellipse,the point $P(m_1, m_2)$:

If tangents are drawn from any point on the circle $x^2+y^2=25$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$,then the angle between the tangents is

Let $(h, k)$ lie on the circle $C: x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac{5}{e^2}$ is equal to . . . . . . .

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

Let $A, B,$ and $C$ be three points on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$. The line joining $A$ and $C$ is parallel to the $x$-axis,and $B$ is the endpoint of the minor axis whose ordinate is positive. Find the maximum area of $\Delta ABC$.