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# Conics - Ellipse - Foci

Suppose $$F$$ and $$F'$$ are the foci of the ellipse $$\displaystyle{\frac{x^2}{121}+\frac{y^2}{64}=1}.$$ If $$P$$ is a point on the ellipse and $$\lvert \overline{FP} \rvert$$ denotes the length of $$\overline{FP},$$ what the minimum value of $$\lvert \overline{FP} \rvert^2 + \lvert \overline{F'P} \rvert^2?$$

The above diagram is an ellipse-shaped track. The distance between the two foci $$F$$ 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 $$\triangle PF'F?$$

For 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 $$PA$$.

In the above diagram, $$F$$ 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 $$a^2+b^2?$$

What is the equation of the circle centered at $$(3,0)$$ which passes through the foci of the ellipse $$\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1?$$

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