Introduction to Conic Sections

Learn what conic sections are and how slicing a cone creates circles, ellipses, parabolas, and hyperbolas.

Advanced25 minLesson
Printable worksheet

Definition

Conic sections are curves obtained by slicing a double-napped cone (two cones placed tip-to-tip) with a flat plane at different angles.
There are four types of conic sections:
  1. 1.Circle: The plane cuts perpendicular to the cone's axis
  2. 2.Ellipse: The plane cuts at an angle, but doesn't intersect the base
  3. 3.Parabola: The plane is parallel to the slant of the cone
  4. 4.Hyperbola: The plane cuts through both cones
Each conic section can also be defined as the set of all points satisfying a specific distance relationship.

Try it now

What type of conic section is x2+y2=25x^2 + y^2 = 25?

Worked Examples

Identify the conic section: x2+y2=16x^2 + y^2 = 16
1

Look at the equation structure

Both x2x^2 and y2y^2 have coefficient 1 → Equal coefficients
2

Check the operation between terms

Addition: x2+y2x^2 + y^2 → Sum of squares
3

Identify the conic

Equal coefficients + addition = circle
4

Find the radius

r2=16r^2 = 16, so r=4r = 4 → Radius = 4

Common Mistakes

Confusing ellipses and circles

Why it's wrong: Both use addition of squared terms, but circles have equal coefficients while ellipses have different coefficients.

Correct: Check if the coefficients (or denominators in standard form) are equal. Equal = circle, different = ellipse.

Forgetting that hyperbolas involve subtraction

Why it's wrong: The negative sign is crucial. Ellipses use ++ between terms, hyperbolas use −-.

Correct: Look for the minus sign: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 is a hyperbola.

Not recognizing parabolas in vertex form

Why it's wrong: Equations like y=a(x−h)2+ky = a(x-h)^2 + k or x=a(y−k)2+hx = a(y-k)^2 + h are parabolas but look different.

Correct: If only one variable is squared, it's a parabola, regardless of the form.

Interactive Visual

Function graph

-6-4-20246-4-2024xyC(0, 0)

Tap a name to show or hide its graph.

  • Center: (0, 0)

Function graph

-10-50510-505xy

Tap or hover over the graph to read off values.

Tap a name to show or hide its graph.

Asymptote
  • 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 = 2x^2

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

17 problems
Problem 1 of 17
Easy
What type of conic section is x2+y2=25x^2 + y^2 = 25?

Why It Matters

Conic sections appear throughout science and engineering:
  • Astronomy: Planets orbit the Sun in ellipses (Kepler's First Law)
  • Physics: Projectiles follow parabolic paths
  • Architecture: Parabolic arches distribute weight efficiently
  • Communications: Satellite dishes use parabolic reflectors to focus signals
  • Navigation: GPS and LORAN systems use hyperbolic positioning
Understanding conics is essential for advanced mathematics, physics, and engineering.

Real World Applications

Planetary Orbits (Ellipses)

Johannes Kepler discovered that planets orbit the Sun in elliptical paths with the Sun at one focus.

Example:

Earth's orbit is nearly circular but slightly elliptical. At its closest (perihelion), Earth is about 147 million km from the Sun; at its farthest (aphelion), about 152 million km.

1Try It Yourself

A comet orbits the Sun in an ellipse. The equation of its orbit is x2100+y236=1\frac{x^2}{100} + \frac{y^2}{36} = 1 (in millions of km).

What is the length of the major axis?

Step 1: Write the mathematical expression

The major axis is 2a2a where a2=100a^2 = 100:

Satellite Dishes (Parabolas)

Satellite dishes are parabolic reflectors that focus incoming signals to a single point called the focus.

Example:

When radio waves hit a parabolic dish, they all reflect to the focus where the receiver is located, amplifying weak signals.

2Try It Yourself

A satellite dish follows the curve y=18x2y = \frac{1}{8}x^2.

If the focus is at (0,p)(0, p) where y=14px2y = \frac{1}{4p}x^2, find the focus.

Step 1: Write the mathematical expression

Compare: 14p=18\frac{1}{4p} = \frac{1}{8}, solve for pp:

Key Takeaways

  • 1Conic sections are curves formed by slicing a double cone with a plane
  • 2The four types are: circle, ellipse, parabola, and hyperbola
  • 3Circles have equal coefficients: x2+y2=r2x^2 + y^2 = r^2
  • 4Ellipses have addition with different denominators: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
  • 5Parabolas have only one squared variable: y=ax2y = ax^2 or x=ay2x = ay^2
  • 6Hyperbolas have subtraction: x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Frequently Asked Questions

They are called conic sections because each curve is formed by the intersection (section) of a plane and a cone. The word 'conic' comes from the Greek 'konikos', meaning 'of a cone'.
They are called conic sections because each curve is formed by the intersection (section) of a plane and a cone. The word 'conic' comes from the Greek 'konikos', meaning 'of a cone'.
A circle is an ellipse where both axes are equal (a=ba = b). When you stretch or compress a circle in one direction, it becomes an ellipse.
Check these: (1) One squared variable = parabola, (2) Both squared with same coefficient and ++ = circle, (3) Both squared with different coefficients and ++ = ellipse, (4) Both squared with −- = hyperbola.

Glossary

Conic section
A curve formed by the intersection of a plane and a double cone
Circle
The set of all points equidistant from a fixed center point
Ellipse
The set of all points where the sum of distances to two foci is constant
Parabola
The set of all points equidistant from a focus point and a directrix line
Hyperbola
The set of all points where the difference of distances to two foci is constant
Focus (plural: foci)
A special point used to define conic sections
Directrix
A line used with the focus to define a parabola
Semi-major axis
Half the length of the longest diameter of an ellipse
Semi-minor axis
Half the length of the shortest diameter of an ellipse

Formula Card

Circle

x2+y2=r2x^2 + y^2 = r^2

Center at origin, radius $r$

Ellipse

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Centered at origin

Parabola (vertical)

y=ax2y = ax^2

Opens up if $a > 0$, down if $a < 0$

Parabola (horizontal)

x=ay2x = ay^2

Opens right if $a > 0$, left if $a < 0$

Hyperbola (horizontal)

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Opens left/right

Hyperbola (vertical)

y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Opens up/down

Today's challengeOne problem, three tries

More in This Topic

Introduction to Ellipses

Advanced25 min

Parabolas in Vertex Form

Advanced25 min

Circles in Standard Form

Advanced25 min

Ellipses in Standard Form

Advanced25 min