The Main Shapesknown As Conics Are

Conics

The main shapesknown as conics are:

Parabola (also known as quadratics as looked at in the last lesson)

Ellipse (including the circle which is a special case)

Hyperbola (including the rectangular hyperbola which is a special case)

Some of them include a term as well as (or instead of) an term

You can recognise a conic from its general equation. The general equation tells you enough about the shape to enable you to sketch it.

The general equations are as follows:

Circle:centre = (,) and radius=

Ellipse:-intercepts =

-intercepts =

Hyperbola:

Points to note:

Circle:If and are both 0, the equation becomes (i.e. a circle with centre at (0, 0) and radius r.)

To find the centre and the radius, the coefficients of and must = 1.

Ellipse:The equation must = 1. (i.e. you need 1 on the right hand side).

You need squared numbers on the bottom of each fraction.

The squared number beneath tells us the -intercepts and the squared number beneath tells us the -intercepts.

Hyperbola:The equation must = 1. (i.e. you need 1 on the right hand side).

You need squared numbers on the bottom of each fraction.

The squared number beneath tells us where to draw the vertical edges of the rectangle and the squared number beneath tells us where to draw the horizontal edges of the rectangle.

Circle

Examples:

Sketch the circle with equation

From the equation we see that the centre of the circle is at and the radius is 4 (because )

Sketch the circle with equation

We need the coefficients of and to equal 1 so we divide the equation by 3.

Starting equation:

÷3:

We now have:

Now we can see that the centre of the circle is at and the radius is 2

Ellipse

Examples:

Sketch the ellipse with equation

We need squared numbers on the bottom of each fraction so we let’s rewrite the equation as:

It’s now easy to see that the ellipse crosses the-axis at and the -axis at

Sketch the ellipse with equation

The equation has to equal 1 so we divide the whole equation by 32.

We now have: (we can rewrite this as: )

From the equation we see that the ellipse crosses the -axis at and the -axis at

Hyperbola

Examples:

Sketch the hyperbola with equation

We can rewrite the equation as:

From the equation we see that we need to draw a rectangle from along the -axis and from along the -axis. We then use the diagonals of the rectangle as asymptotes for the hyperbola. (It might be a good idea to draw the rectangle and it’s diagonals in pencil so you can erase them when you have used them as a guide to draw your hyperbola).

Sketch the hyperbola with equation

The equation has to equal 1 so we need to divide the whole equation by 8.

We now have: We can rewritethis as:

This time draw your rectangle from along the -axis and from along the -axis.