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 curvesβ€”circle, ellipse, parabola, and hyperbolaβ€”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 y2βˆ’8x=0y^2 - 8x = 0.

Example 2

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

Example 3

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

Example 4

easy
What conic has eccentricity e=0e=0?

Example 5

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

Example 6

medium
What degenerate conic does x2βˆ’y2=0x^2 - y^2 = 0 represent?

Example 7

hard
For what values of kk is kx2+(kβˆ’2)y2=1kx^2 + (k-2)y^2 = 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=x2βˆ’3y = x^2 - 3.

Example 10

medium
Classify x2+y2+4xβˆ’6y+4=0x^2 + y^2 + 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 3x2βˆ’2xy+3y2βˆ’8=03x^2 - 2xy + 3y^2 - 8 = 0.

Example 13

easy
What conic has eccentricity exactly e=1e=1?

Example 14

medium
Use B2βˆ’4ACB^2-4AC to classify 4x2+4xy+y2βˆ’8=04x^2 + 4xy + y^2 - 8 = 0.

Example 15

challenge
For what Ξ»\lambda is x2+Ξ»xy+y2=1x^2 + \lambda xy + y^2 = 1 a pair of lines (degenerate)?

Example 16

challenge
A conic has eccentricity e=12e=\tfrac{1}{2} and focus at the origin with directrix x=4x=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+4y2βˆ’6=0x^2 + 4xy + 4y^2 - 6 = 0 via the discriminant.

Example 19

medium
What does x2+y2=0x^2+y^2=0 represent?

Example 20

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