# 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

**Slope-intercept Form**

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

y=mx+b

**Point-slope form**

The equation of a line passing through a point (*x*_{1},* y*_{1}) with slope *m*:

y−y_{1}=m(x−x_{1})

**General Form of a Straight Line **

`Ax + By + C = 0`

**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.

## Circle

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

x^{2}+y^{2}=r^{2}

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

(

x−h)^{2}+ (y−k)^{2}=r^{2}

**General Form of the Circle**

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

x^{2}+y^{2}+Dx+Ey+F= 0

**Formal Definition**

**Conic Section**

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 with Vertical Axis **

A parabola with focal distance* p* has equation:

x^{2}= 4py

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

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

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

(x−h)^{2}= 4p(y−k)

**Parabola with Horizontal Axis **

In this case, we have the *relation:*

y^{2}= 4px

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

(

y−k)^{2}= 4p(x−h)

**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.

For background and examples, see Parabola.

## Ellipse

**Horizontal Major Axis**

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)`.

An ellipse 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**

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).

For background and examples, see Ellipse.

## Hyperbola

**North-south Opening **

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`

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 **

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`

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.

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.