Conic sections - summary

This is a summary of the first 5 topics in this chapter: straight line, circle, parabola, ellipse and hyperbola.

Don't miss the 3D interactive graph, where you can explore these conic sections by slicing a double cone.

Straight Line

y = mx + b graph

Slope-intercept Form

The equation of a line with slope m and y-intercept b is given by:

y = mx + b

point-slope graph

Point-slope form

The equation of a line passing through a point (x1, y1) with slope m:

y y1 = m(xx1)

General Form of a Straight Line

`Ax + By + C = 0`
conic - straight line

Conic Section

If we slice the double cone by a plane just touching one edge of the double cone, the intersection is a straight line, as shown.

 

For background and examples, see Straight Line.

Continues below

Circle

circle center origin

The circle with centre (0, 0) and radius r has the equation:

x2 + y2 = r2

circle center (h,k)

The circle with centre (h, k) and radius r has the equation:

(xh)2 + (yk)2 = r2

General Form of the Circle

An equation which can be written in the following form (with constants D, E, F) represents a circle:

x2 + y2 + Dx + Ey + F = 0

Formal Definition A circle is the locus of points that are equidistant from a fixed point (the center).

Conic Section

conic - circle

If we slice one of the cones with a plane at right angles to the axis of the cone, the shape formed is a circle.

For background and examples, see Circle.

Parabola

parabola

Parabola with Vertical Axis

A parabola with focal distance p has equation:

x2 = 4py

In more familiar form, with "y = " on the left, we can write this as:

`y=x^2/(4p)`

parabola vertex (h,k)

If the axis of a parabola is vertical, and the vertex is at (h, k), we have

(xh)2 = 4p(yk)
parabola horizontal axis

Parabola with Horizontal Axis

In this case, we have the relation:

y2 = 4px

shifted parabola horizontal axis

If the axis of a parabola is horizontal, and the vertex is at (h, k), the equation becomes

(yk)2 = 4p(xh)

Formal Definition A parabola is the locus of points that are equidistant from a point (the focus) and a line (the directrix).

Conic Section

If we slice a cone parallel to the slant edge of the cone, the resulting shape is a parabola, as shown.

conic parabola conic parabola

For background and examples, see Parabola.

Ellipse

Horizontal Major Axis

ellipse graph

The equation for an ellipse with a horizontal major axis and center (0,0) is given by:

`x^2/a^2+y^2/b^2=1`

The foci (plural of 'focus') of the ellipse (with horizontal major axis) are at `(-c,0)` and `(c,0)`, where c is given by:

`c=sqrt(a^2-b^2`

The vertices of an ellipse are at `(-a, 0)` and `(a, 0)`.

shifted ellipse

A parabola with horizontal major axis and with center at (h, k) is given by:

`((x-h)^2)/a^2+((y-k)^2)/b^2=1`

Vertical Major Axis

ellipse vertical axis

If the major axis is vertical, then the formula becomes:

`x^2/b^2+y^2/a^2=1`

We always choose our a and b such that a > b.

Formal Definition An ellipse is the locus of points whereby the sum of the distances from 2 fixed points (the foci) is constant..

Conic Section

 

When we slice one of the cones at an angle to the sides of the cone, we get an ellipse, as seen in the view from the top (at right).

conic ellipse conic ellipse

For background and examples, see Ellipse.

Hyperbola

North-south Opening

hyperbola

For a north-south opening hyperbola:

`y^2/a^2-x^2/b^2=1`

The slopes of the asymptotes are given by:

`+-a/b`

hyperbola shifted

For a "north-south" opening hyperbola with centre (h, k), we have:

`((y-k)^2)/a^2-((x-h)^2)/b^2=1`

East-west Opening

east west hyperbola

For an east-west opening hyperbola:

`x^2/a^2-y^2/b^2=1`

The slopes of the asymptotes are given by:

`+-b/a`

 

shifted east-west hyperbola

For an "east-west" opening hyperbola with centre (h, k), we have:

`((x-h)^2)/a^2-((y-k)^2)/b^2=1`

Formal Definition A hyperbola is the locus of points where the difference in the distance to two fixed foci is constant.

General Form of a Hyperbola

`Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`

(such that `B^2>4AC`)

Conic Section

 

When we slice our double cone such that the plane passes througn both cones, we get a hyperbola, as shown.

hyperbola conic

For background and examples, see Hyperbola.

Also, don't miss the 3D interactive graph, where you can explore these conic sections by slicing a double cone.

top

Search IntMath, blog and Forum

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Math Lessons on DVD

 

Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.