ConicsWhat are they? What do they look like? What are their equations?

The Word Conic Comes from the shape of a cone and the four examples of “conic sections” come from taking certain slices out of the 3D cone and observing them

from a 2D perspective. This lets us see certain shapes that we can map onto a graph with x and y axis that we can translate and reflect.

There are four conic Sections that can be taken from a cone and they are:

Ellipses Hyperbolae Circles Parabolas

By Anderson McCammont 12JA

x

y

EllipsesAn Ellipse is a shape that is similar to the appearance of an oval. It has the equation of:

x 2 y 2

a 2 b 2

In this example to the right hand side, we can see that there are two values on the graph. The value of a is 2 and the value of b is 1. We know this because as it is “x squared over a squared”, a is paired with x and so the value that lies on the ellipse and on the x axis is the value of a. The value of b is 1 because, exactly the same as the reason previously stated, the value that lies on the ellipse and y axis is the value of b.

2

1

Hyperbolae

x

y

x 2 y 2

a 2 b 2

A Hyperbola has a similar equation to that of an ellipse the only difference being that the sign is a minus instead of an add. This effects theConic so that it looks like the image shown below.

The value of a that is used in the hyperbolic equation is where the hyperbola itself crosses the x axis. This is a and –a. Using the equations of y=b/a x and y=-b/a x, you can sub in the equations of the lines of the asymptotes and then find b from that which finally lets you write the hyperbola in the form of an equation seen below.

(x-a)2+(y-b)2=r2

Circles

Circles are very different to the other conics as their equation involves radii.In the case of the other conics, the values of a and b were determined by

The location of the points of intersection, this time the value of a and b are given byFinding out the centre of the circle in the form (x,y). The x value in this co-ordinate is nowThe value of a and the y value is the value of b. R on the other hand is as stated earlier the

value of the radius, squared. Ill give an example:

The centre of a circle is:

(3,4) with radius 5, write this in the form of the equation of a circle.

Using the equation setup and the information given, we can rewrite the data as:

(x-3) 2 + (y-4) 2 =52

This finally becomes: (x-3) 2 + (y-4) 2 =25

Parabolas

x

y

This is the graph of y=x2-6x+5

ax2+bx+c=0A Parabola is a graph that crosses the x axis at certain points that can be determined by factorising the equation given, it will occasionally have no points of intersection but for this example it has two. y=x2-6x+5Can be factorised into (x-5) and (x-1). In order to find out where is crosses you need to make the factors equal to zero.X-5=0 x-1=0 Solving gives x values of 1 and 5. the point at which the curve crosses the y axis is determined by the arbitrary constant in the equation, in this case, 5.

5

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