Geometry
# Conic Sections

$F$ and $F'$ is $8,$ and the sum of the distances between any point $P$ on the inner side of the track and $F$ and $F'$ is $32.$ What is the maximum area of $\triangle PF'F?$

The above diagram is an ellipse-shaped track. The distance between the two fociFor an ellipse $E$ with foci $A=(2,0)$ and $B=(10,0),$ every point $P$ on $E$ satisfies $\lvert PA \rvert + \lvert PB \rvert = 30.$ What is the equation of ellipse $E?$

Note: $\lvert PA \rvert$ denotes the length of line segment $PA$.

$F$ and $F'$ are the foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $A$ and $A'$ are the end points of the major axis of the ellipse. If the area of triangle $\triangle A'PA$ is double the area of triangle $\triangle F'PF$ and the perimeter of triangle $\triangle F'PF$ is $66,$ what is the value of $a^2+b^2?$

In the above diagram,