Identifying Conic Sections

Learn to distinguish between circles, ellipses, parabolas, and hyperbolas from their equations.

Advanced25 minLesson
Printable worksheet

Definition

A conic section is a curve formed by the intersection of a plane with a double cone. The four types are:
  • Circle: All points equidistant from a center
  • Ellipse: Stretched circle with two focal points
  • Parabola: U-shaped curve with one focus and directrix
  • Hyperbola: Two separate curved branches

Identifying from Standard Form

ConicStandard FormKey Features
Circle(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2Equal coefficients, both positive
Ellipse(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1Different denominators, both positive
Parabolay=a(x−h)2+ky = a(x-h)^2 + k or x=a(y−k)2+hx = a(y-k)^2 + hOnly one variable is squared
Hyperbola(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1Subtraction between terms

Identifying from General Form

The general form is: Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
When B=0B = 0 (no xyxy term):
  • Circle: A=CA = C (same coefficients)
  • Ellipse: AA and CC have the same sign but A≠CA \neq C
  • Parabola: Either A=0A = 0 or C=0C = 0 (only one squared term)
  • Hyperbola: AA and CC have opposite signs

Try it now

Identify the conic section: x2+y2=25x^2 + y^2 = 25

Worked Examples

Identify the conic section: x2+y2−6x+4y−12=0x^2 + y^2 - 6x + 4y - 12 = 0
1

Check for an xy term

There is no xyxy term, so B=0B = 0 → No rotation needed
2

Identify coefficients of squared terms

A=1A = 1 (coefficient of x2x^2) and C=1C = 1 (coefficient of y2y^2) → A=1A = 1, C=1C = 1
3

Compare A and C

A=C=1A = C = 1 (equal and both positive) → This is a circle
4

Verify by completing the square

(x−3)2+(y+2)2=25(x-3)^2 + (y+2)^2 = 25 → Circle with center (3,−2)(3, -2) and radius 55

Common Mistakes

Confusing an ellipse with a circle when coefficients look similar

Why it's wrong: Students may not notice that coefficients like 4x2+9y24x^2 + 9y^2 mean different denominators in standard form.

Correct: For a circle, coefficients must be exactly equal (A=CA = C). If 4x2+9y2=364x^2 + 9y^2 = 36, dividing gives x29+y24=1\frac{x^2}{9} + \frac{y^2}{4} = 1 (ellipse, not circle).

Forgetting that negative coefficients indicate a hyperbola

Why it's wrong: Students see x2−y2x^2 - y^2 and don't recognize the subtraction pattern.

Correct: When AA and CC have opposite signs (one positive, one negative), it's always a hyperbola.

Not recognizing a parabola when one squared term is missing

Why it's wrong: The equation y=3x2+2x−5y = 3x^2 + 2x - 5 doesn't look like a typical conic form.

Correct: If only x2x^2 or only y2y^2 appears (not both), it's a parabola regardless of how the equation is written.

Dividing incorrectly when converting to standard form

Why it's wrong: Dividing 4x2+y2=164x^2 + y^2 = 16 by 16 gives x24+y216\frac{x^2}{4} + \frac{y^2}{16}, not 4x216\frac{4x^2}{16}.

Correct: When dividing 4x24x^2 by 16, simplify: 4x216=x24\frac{4x^2}{16} = \frac{x^2}{4}.

Interactive Visual

Function graph

-6-4-20246810-8-6-4-2024xyC(3, -2)

Tap a name to show or hide its graph.

  • Center: (3, -2)

Function graph

-10-50510-10-505xyFVx = 0

Tap a name to show or hide its graph.

DirectrixAsymptote
  • Focus: (4, -3)
  • Directrix: x = 0
  • Vertex: (2, -3)
  • Asymptotes: y = 0.75x, y = -0.75x

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2*sqrt(1 - x^2/9)

y = -2*sqrt(1 - x^2/9)

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Identify the conic section: x2+y2=25x^2 + y^2 = 25

Why It Matters

Identifying conic sections is essential for:
  • Physics: Planetary orbits are ellipses, projectile paths are parabolas, and hyperbolas describe certain particle trajectories
  • Engineering: Satellite dishes and car headlights use parabolic reflectors; suspension bridge cables form parabolas
  • Architecture: The Colosseum in Rome has an elliptical shape; many modern buildings feature hyperbolic structures
  • Astronomy: Understanding orbital mechanics requires recognizing conic sections
  • Navigation: GPS and radar systems use properties of conics for positioning
Being able to quickly identify the type of conic from an equation saves time and helps you choose the right approach for graphing or solving problems.

Real World Applications

Satellite Dish Design

Satellite dishes are parabolic reflectors. Engineers need to identify the parabola equation to calculate the focal point where the receiver should be placed.

Example:

A dish follows y=0.0625x2y = 0.0625x^2. Since only xx is squared, it's a parabola. The focal length is 14a=14(0.0625)=4\frac{1}{4a} = \frac{1}{4(0.0625)} = 4 meters.

1Try It Yourself

A radio telescope dish has equation x2+y2−100=0x^2 + y^2 - 100 = 0.

Is this a parabolic dish or circular dish?

Step 1: Choose the right answer

Check if both variables are squared equally. Which conic is it?

Orbital Mechanics

Planets orbit in ellipses, comets can follow parabolic or hyperbolic paths. Identifying the conic helps predict the object's trajectory.

Example:

An object's orbit satisfies x2100+y264=1\frac{x^2}{100} + \frac{y^2}{64} = 1. Both terms positive with different denominators means ellipse - this is a bound orbit.

2Try It Yourself

A comet's path is modeled by x2−y24=1x^2 - \frac{y^2}{4} = 1.

Will this comet return to our solar system?

Step 1: Choose the right answer

Identify the conic type:

Architectural Design

Architects use conic sections in building design. Elliptical rooms create whispering galleries, while hyperbolic cooling towers are structurally efficient.

Example:

A building footprint follows 4x2+4y2=1004x^2 + 4y^2 = 100. Since A=C=4A = C = 4, this is a circle with radius 5 meters.

3Try It Yourself

An amphitheater is designed with equation x2400+y2225=1\frac{x^2}{400} + \frac{y^2}{225} = 1.

What shape is the amphitheater?

Step 1: Choose the right answer

Analyze the equation form:

Key Takeaways

  • 1Conic sections are circles, ellipses, parabolas, and hyperbolas
  • 2Circle: A=CA = C (equal coefficients, both positive)
  • 3Ellipse: AA and CC same sign, but A≠CA \neq C (different positive coefficients)
  • 4Parabola: Only one variable is squared (A=0A = 0 or C=0C = 0)
  • 5Hyperbola: AA and CC have opposite signs (subtraction between squared terms)
  • 6Always check for xyxy terms - if present, the conic is rotated

Frequently Asked Questions

An xyxy term (B≠0B \neq 0) indicates a rotated conic. To identify it, calculate the discriminant B2−4ACB^2 - 4AC: if negative, it's an ellipse or circle; if zero, a parabola; if positive, a hyperbola.
An xyxy term (B≠0B \neq 0) indicates a rotated conic. To identify it, calculate the discriminant B2−4ACB^2 - 4AC: if negative, it's an ellipse or circle; if zero, a parabola; if positive, a hyperbola.
Yes! Some equations like x2+y2=−1x^2 + y^2 = -1 have no real solutions (imaginary circle). These are called degenerate conics.
Use this memory trick: Circle = Coefficients equal; Ellipse = Equal signs, unequal values; Parabola = Partially squared (one variable); Hyperbola = Has opposite signs.

Glossary

Conic Section
A curve formed by intersecting a plane with a double cone: circle, ellipse, parabola, or hyperbola
General Form
The equation Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Standard Form
The simplified form of a conic equation that reveals its center, vertices, or other key features
Discriminant
For conics: B2−4ACB^2 - 4AC, used to identify rotated conic sections
Degenerate Conic
A conic that reduces to a point, line, or pair of lines instead of a curve
Today's challengeOne problem, three tries

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Introduction to Ellipses

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Parabolas in Vertex Form

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Circles in Standard Form

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