Conic Sections Overview Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Conic Sections Overview.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Circle, ellipse, parabola, hyperbola are all cuts of a double cone, sorted by the discriminant B2−4ACB^2-4AC.

Common stuck point: The procedure for conic sections overview is the easy part; the trap is reading B2−4ACB^2-4AC as a root count. Asking "Am I deciding the TYPE of conic from a general equation rather than analyzing one specific known conic?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I deciding the TYPE of conic from a general equation rather than analyzing one specific known conic?

Worked Examples

Example 1

easy
Identify the type of conic section: x29+y29=1\frac{x^2}{9} + \frac{y^2}{9} = 1.

Answer

Circle with radius 3\text{Circle with radius } 3

First step

1
The equation has the form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 with both terms positive (addition).

Full solution

  1. 2
    Since a2=b2=9a^2 = b^2 = 9, we have a=b=3a = b = 3.
  2. 3
    An ellipse with a=ba = b is a circle of radius 33 centered at the origin.
A circle is a special case of an ellipse where both semi-axes are equal (a=ba = b). The four conic sections — circle, ellipse, parabola, and hyperbola — are all cross-sections of a cone, distinguished by the angle at which the cutting plane intersects the cone.

Example 2

medium
Classify the conic given by 4x2−9y2+16x+18y−29=04x^2 - 9y^2 + 16x + 18y - 29 = 0.

Example 3

challenge
Show the polar equation r=21−cos⁡θr = \dfrac{2}{1-\cos\theta} describes a parabola.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Classify each equation: (a) x2+y2−6x+2y+1=0x^2 + y^2 - 6x + 2y + 1 = 0, (b) x2−4y+8=0x^2 - 4y + 8 = 0, (c) 9x2+4y2=369x^2 + 4y^2 = 36.

Example 2

hard
The general equation 2x2+2y2+Bxy−8=02x^2 + 2y^2 + Bxy - 8 = 0 represents a circle only when BB takes a specific value. Find that value.

Example 3

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

Example 4

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

Example 5

easy
What conic has eccentricity e>1e>1?

Example 6

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

Example 7

easy
Which conic results from slicing a cone parallel to its base?

Example 8

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

Example 9

easy
In Ax2+Cy2+…=0Ax^2+Cy^2+\ldots=0 with B=0B=0, what makes it a circle?

Example 10

easy
What is the conic discriminant used to classify Ax2+Bxy+Cy2+…=0Ax^2+Bxy+Cy^2+\ldots=0?

Example 11

medium
Classify 4x2+9y2−36=04x^2 + 9y^2 - 36 = 0.

Example 12

medium
Classify x2−4y2−16=0x^2 - 4y^2 - 16 = 0.

Example 13

medium
Classify y2−8x=0y^2 - 8x = 0.

Example 14

medium
Use B2−4ACB^2-4AC to classify 3x2+5xy+2y2=13x^2 + 5xy + 2y^2 = 1.

Example 15

medium
Classify x2+4xy+4y2−6=0x^2 + 4xy + 4y^2 - 6 = 0 via the discriminant.

Example 16

medium
Classify 2x2+3y2+xy−5=02x^2 + 3y^2 + xy - 5 = 0.

Example 17

medium
What degenerate conic does x2−y2=0x^2 - y^2 = 0 represent?

Example 18

challenge
For what kk is x2+ky2=1x^2 + ky^2 = 1 an ellipse, a circle, and a hyperbola?

Example 19

challenge
A conic has e=1e=1 and focus (1,0)(1,0) with vertex at origin. Identify it and find pp.

Example 20

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

Example 21

medium
Classify x2+y2−6x+4=0x^2 + y^2 - 6x + 4 = 0.

Example 22

medium
Classify 9x2−4y2+18x−16y=439x^2 - 4y^2 + 18x - 16y = 43 by its squared terms.

Example 23

easy
Classify x2+y2=25x^2 + y^2 = 25.

Example 24

easy
Classify y=x2−3y = x^2 - 3.

Example 25

easy
What is the eccentricity of x225+y225=1\frac{x^2}{25}+\frac{y^2}{25}=1?

Example 26

medium
Use B2−4ACB^2-4AC to classify x2−xy+y2=4x^2 - xy + y^2 = 4.

Example 27

medium
Use B2−4ACB^2-4AC to classify 4x2+4xy+y2−8=04x^2 + 4xy + y^2 - 8 = 0.

Example 28

medium
Classify 25x2+16y2=40025x^2 + 16y^2 = 400.

Example 29

medium
Classify y24−x29=1\frac{y^2}{4} - \frac{x^2}{9} = 1.

Example 30

medium
Classify x2+y2+4x−6y+4=0x^2 + y^2 + 4x - 6y + 4 = 0.

Example 31

medium
Classify x2−6x−y+10=0x^2 - 6x - y + 10 = 0.

Example 32

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

Example 33

medium
What does x2+y2=−1x^2+y^2=-1 represent?

Example 34

medium
Use the discriminant to classify 3x2−2xy+3y2−8=03x^2 - 2xy + 3y^2 - 8 = 0.

Example 35

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

Example 36

hard
Classify 9x2−4y2−18x+8y−31=09x^2 - 4y^2 - 18x + 8y - 31 = 0 by grouping.

Example 37

hard
Classify 4x2+9y2+16x−18y−11=04x^2 + 9y^2 + 16x - 18y - 11 = 0 and find its center.

Example 38

hard
Identify the conic with focus (0,2)(0,2) and directrix y=−2y=-2.

Example 39

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

Example 40

hard
Determine all values of kk for which x2+y2+4x−2y+k=0x^2 + y^2 + 4x - 2y + k = 0 represents a real circle.

Example 41

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

Example 42

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.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

equation of circleellipsehyperbolaparabola focus directrix