Practice Conic Sections Overview in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The four curves—circle, ellipse, parabola, and hyperbola—obtained by slicing a double cone with a plane at different angles.

Imagine a flashlight shining on a wall. Straight on: circle. Tilted slightly: ellipse. Tilted to match the cone's edge: parabola. Tilted past the edge: hyperbola. All four shapes come from the same geometric object (a cone), just viewed from different angles.

Showing a random 20 of 50 problems.

Example 1

medium
Classify y2−8x=0.

Example 2

challenge
Order circle, ellipse, parabola, hyperbola by increasing eccentricity, with values.

Example 3

hard
A conic has foci (±3,0) and passes through (5,0). Identify it and write the equation.

Example 4

easy
What conic has eccentricity e=0?

Example 5

easy
What range of eccentricity defines an ellipse (non-circular)?

Example 6

medium
What degenerate conic does x2−y2=0 represent?

Example 7

hard
For what values of k is kx2+(k−2)y2=1 an ellipse (not a circle)?

Example 8

easy
Which conic has two foci and the absolute difference of distances to them constant?

Example 9

easy
Classify y=x2−3.

Example 10

medium
Classify x2+y2+4x−6y+4=0.

Example 11

easy
Which conic has two foci and the sum of distances to them constant?

Example 12

medium
Use the discriminant to classify 3x2−2xy+3y2−8=0.

Example 13

easy
What conic has eccentricity exactly e=1?

Example 14

medium
Use B2−4AC to classify 4x2+4xy+y2−8=0.

Example 15

challenge
For what λ is x2+λxy+y2=1 a pair of lines (degenerate)?

Example 16

challenge
A conic has eccentricity e=12 and focus at the origin with directrix x=4. Identify the conic and find its equation in polar form.

Example 17

medium
Which slice of a double cone produces a hyperbola?

Example 18

medium
Classify x2+4xy+4y2−6=0 via the discriminant.

Example 19

medium
What does x2+y2=0 represent?

Example 20

easy
Which conic results from a plane parallel to the cone's slant edge?
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