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 - General

Let $(m,n)$ be the point at which a line is tangent to the ellipse $\displaystyle{\frac{x^2}{16}+\frac{y^2}{100}=1}.$ If this line has $x$ -intercept $A=(a,0)$ and $y$ -intercept $B=(0,b),$ where $a$ and $b$ are positive, what is the minimum value of $ab?$

78 80 82 84

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by Brilliant Staff

The ellipse $\displaystyle{\frac{x^2}{8}+\frac{y^2}{17}=1}$ and the line $y=2x - k$ intersect at exactly two distinct points. What is the range of the constant $k?$

-\sqrt{{47}}<k< \sqrt{{47}}

-\sqrt{{53}}<k< \sqrt{{53}}

-\sqrt{{49}}<k< \sqrt{{49}}

-\sqrt{{51}}<k< \sqrt{{51}}

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by Brilliant Staff

Find the equation of the tangent line to the ellipse $1x^2+y^2=53$ at the point $(2,7)$ on the ellipse.

2x-7y=53

2x-7y=51

2x+7y=51

2x+7y=53

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by Brilliant Staff

In the above diagram, a rectangle $ABCD$ is inscribed in the ellipse $\displaystyle{\frac{x^2}{36}+\frac{y^2}{100}=1}.$ What is the maximum area of rectangle $ABCD?$

Note: The above diagram is not drawn to scale.

122 118 124 120

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by Brilliant Staff

Find the maximum area of the rectangle which is inscribed in the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{2^{2}} = 1.$

40 20 10 30

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by Brilliant Staff

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