Excel in math and science.

Join using Facebook Join using Google Join using email

Forgot password? New user? Sign up

Existing user? Log in

Your answer seems reasonable. Find out if you're right!

That seems reasonable. Find out if you're right!

Join using Facebook Join using Google Join using email

Forgot password? New user? Sign up

Existing user? Log in

Geometry

Conic Sections

Concept Quizzes

Conics - Circle Standard Equation
Conics - Circle General Equation
Conics - Parabola - General
Conics - Parabola - Focus and Directrix
Conics - Ellipse - General
Conics - Ellipse - Foci
Conics - Hyperbola
Conics - Discriminant
Conic Sections - Problem Solving

Challenge Quizzes

Conic Sections: Level 2 Challenges
Conic Sections: Level 3 Challenges
Conic Sections: Level 5 Challenges

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 $\lvert \overline{FP} \rvert^2 + \lvert \overline{F'P} \rvert^2?$

240 244 246 242

Show explanation

by Brilliant Staff

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 $\triangle PF'F?$

{3}\sqrt{240}

{4}\sqrt{240}

{4}\sqrt{238}

{3}\sqrt{238}

Show explanation

by Brilliant Staff

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 $PA$ .

\frac{(x-{8})^2}{225}+\frac{y^2}{211}=1

\frac{(x-{6})^2}{217}+\frac{y^2}{218}=1

\frac{(x-{6})^2}{225}+\frac{y^2}{209}=1

\frac{(x-{8})^2}{235}+\frac{y^2}{205}=1

Show explanation

by Brilliant Staff

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 $a^2+b^2?$

851 845 849 847

Show explanation

by Brilliant Staff

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

x^{2} + y^{2} - 6x + 5 = 0

x^{2} + y^{2} - 6x - 7 = 0

x^{2} + y^{2} - 6x - 5 = 0

x^{2} + y^{2} - 6x - 25 = 0

Show explanation

by Brilliant Staff