Conic Sections

Conics - Ellipse - Foci


Suppose FF and FF' are the foci of the ellipse x2121+y264=1.\displaystyle{\frac{x^2}{121}+\frac{y^2}{64}=1}. If PP is a point on the ellipse and FP\lvert \overline{FP} \rvert denotes the length of FP,\overline{FP}, what the minimum value of FP2+FP2?\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 FF and FF' is 8,8, and the sum of the distances between any point PP on the inner side of the track and FF and FF' is 32.32. What is the maximum area of PFF?\triangle PF'F?

For an ellipse EE with foci A=(2,0)A=(2,0) and B=(10,0),B=(10,0), every point PP on EE satisfies PA+PB=30. \lvert PA \rvert + \lvert PB \rvert = 30. What is the equation of ellipse E?E?

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

In the above diagram, FF and FF' are the foci of the ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 and AA and AA' are the end points of the major axis of the ellipse. If the area of triangle APA\triangle A'PA is double the area of triangle FPF\triangle F'PF and the perimeter of triangle FPF\triangle F'PF is 66,66, what is the value of a2+b2?a^2+b^2?

What is the equation of the circle centered at (3,0)(3,0) which passes through the foci of the ellipse x29+y216=1?\frac{x^{2}}{9} + \frac{y^{2}}{16} = 1?


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