Conics

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Conic Section diagram

There are 4 graphs that are formed when a cone is sliced by a flat surface. These conic sections are:


Circle Equations

Carnival Ride

The Extreme Orbiter above consists of five rotating arms with three pairs of spinning seats on each arm. The engineer who designed this carnival ride used this equation of a circle which is based on Pythagoras' Rule.

( x – h )2 + ( y – k )2 = r2

where
(h,k) is the centre
r is the radius of the circle.


Example One

Find the coordinates of the centre and the length of the radius of a circle whose equation is:

( x – 3)2 + ( y + 4 )2 = 25.

Then draw this circle on grid paper.

Answer:
Centre = (h,k) = (3, –4)
Radius = 5

Conics Example diagram


Ellipse Equations

Planetary Orbits


( x – h )2 + ( y – k )2 = 1
a2b2

where
(h,k) is the centre
a is half of the major axis
b is half of the minor axis of the ellipse.


Example Two

Find the coordinates of the centre and the lengths of the axes whose equation is:

( x – 3 )2 + ( y + 4 )2 = 1
4925

Answer:
h = 3
k = –4
a = 7
b = 5

Centre = (h,k) = (3, –4)
Length of major axis = 7 × 2 = 14
Length of minor axis = 5 × 2 = 10

Conics Example diagram


Hyperbola Equations

Hyperbolic Light

The light given out by this lamp forms hyperbolas.


Example Three

Draw the graphs of the following hyperbolic equations. What do you notice?

(a)y = 1
x
(b)y = 1
x – 3
(c)y = 5
x – 3
(d)y = 5 + 4
x – 3

Answers:

(a)

Conics Example diagram

(b)

Conics Example diagram

(c)

Conics Example diagram

(d)

Conics Example diagram

Maths Fun - Pizza Pieces

Pizza Cuts

Below are the maximum numbers of pieces that you can get when making straight cuts of pizza.

By drawing a circle or more enjoyably, cutting (and then eating) a pizza, continue cutting and counting. If x is the number of cuts and y is the number of pieces, what is the equation that links these?