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A circle is easy to make:
Draw a curve that is 'radius' away
from a central point.
from a central point.
And so:
All points are the same distance
from the center.
from the center.
In fact the definition of a circle is
Circle: The set of all points on a plane that are a fixed distance from a center.
Circle on a Graph
Let us put a circle of radius 5 on a graph:
Now let's work out exactly where all the points are.
We make a right-angled triangle:
And then use Pythagoras:
x2 + y2 = 52
There are an infinite number of those points, here are some examples:
x | y | x2 + y2 |
---|---|---|
5 | 0 | 52 + 02 = 25 + 0 = 25 |
3 | 4 | 32 + 42 = 9 + 16 = 25 |
0 | 5 | 02 + 52 = 0 + 25 = 25 |
−4 | −3 | (−4)2 + (−3)2 = 16 + 9 = 25 |
0 | −5 | 02 + (−5)2 = 0 + 25 = 25 |
In all cases a point on the circle follows the rule x2 + y2 = radius2
We can use that idea to find a missing value
Example: x value of 2, and a radius of 5
Values we know:22 + y2 = 52
Square root both sides: y = ±√(52 − 22)
y ≈ ±4.58..
(The ± means there are two possible values: one with + the other with −) Yummy ftp 1 10 2 download free.
And here are the two points:
More General Case
Now let us put the center at (a,b)
So the circle is all the points (x,y) that are 'r' away from the center (a,b).
![Waltr Waltr](https://i0.wp.com/4macsoft.com/wp-content/uploads/2019/12/WALTR-2.6.25-Mac-Free-Download.jpg?resize=620%2C418&ssl=1)
Now lets work out where the points are (using a right-angled triangle and Pythagoras):
It is the same idea as before, but we need to subtract a and b:
(x−a)2 + (y−b)2 = r2
And that is the 'Standard Form' for the equation of a circle!
It shows all the important information at a glance: the center (a,b) and the radius r.
Example: A circle with center at (3,4) and a radius of 6:
Start with:
(x−a)2 + (y−b)2 = r2
Put in (a,b) and r:
![Waltr 2 6 25 x 4 Waltr 2 6 25 x 4](https://static.rcgroups.net/forums/attachments/6/9/2/2/0/7/t9873165-111-thumb-piezo-buzzer-specs-500x500_0%5B1%5D.jpg?d=1489672739)
(x−3)2 + (y−4)2 = 62
We can then use our algebra skills to simplify and rearrange that equation, depending on what we need it for.
Try it Yourself
'General Form'
But you may see a circle equation and not know it!
Because it may not be in the neat 'Standard Form' above.
As an example, let us put some values to a, b and r and then expand it
Example: a=1, b=2, r=3:(x−1)2 + (y−2)2 = 32
Waltr 2 6 25 X 4 4
Gather like terms:x2 + y2 − 2x − 4y + 1 + 4 − 9 = 0
And we end up with this:
x2 + y2 − 2x − 4y − 4 = 0
It is a circle equation, but 'in disguise'!
So when you see something like that think 'hmm .. that might be a circle!'
In fact we can write it in 'General Form' by putting constants instead of the numbers:
Note: General Form always has x2 + y2 for the first two terms.
Going From General Form to Standard Form
Now imagine we have an equation in General Form:
x2 + y2 + Ax + By + C = 0
How can we get it into Standard Form like this?
(x−a)2 + (y−b)2 = r2
The answer is to Complete the Square (read about that) twice .. once for x and once for y:
Example: x2 + y2 − 2x − 4y − 4 = 0
Put xs and ys together:(x2 − 2x) + (y2 − 4y) − 4 = 0
Now complete the square for x (take half of the −2, square it, and add to both sides):
(x2 − 2x + (−1)2) + (y2 − 4y) = 4 + (−1)2
And complete the square for y (take half of the −4, square it, and add to both sides):
(x2 − 2x + (−1)2) + (y2 − 4y + (−2)2) = 4 + (−1)2 + (−2)2
Tidy up:
Finally:(x − 1)2 + (y − 2)2 = 32
Waltr 2 6 25 X 4 50
And we have it in Standard Form!
(Note: this used the a=1, b=2, r=3 example from before, so we got it right!)
Unit Circle
If we place the circle center at (0,0) and set the radius to 1 we get:
(x−a)2 + (y−b)2 = r2 (x−0)2 + (y−0)2 = 12 x2 + y2 = 1 Which is the equation of the Unit Circle |
How to Plot a Circle by Hand
1. Plot the center (a,b)
2. Plot 4 points 'radius' away from the center in the up, down, left and right direction
3. Sketch it in!
Example: Plot (x−4)2 + (y−2)2 = 25
The formula for a circle is (x−a)2 + (y−b)2 = r2
So the center is at (4,2)
And r2 is 25, so the radius is √25 = 5
So we can plot:
- The Center: (4,2)
- Up: (4,2+5) = (4,7)
- Down: (4,2−5) = (4,−3)
- Left: (4−5,2) = (−1,2)
- Right: (4+5,2) = (9,2)
Now, just sketch in the circle the best we can!
How to Plot a Circle on the Computer
We need to rearrange the formula so we get 'y='.
We should end up with two equations (top and bottom of circle) that can then be plotted.
Example: Plot (x−4)2 + (y−2)2 = 25
So the center is at (4,2), and the radius is √25 = 5
Waltr 2 6 25 X 4 25
Rearrange to get 'y=':
Move (x−4)2 to the right: (y−2)2 = 25 − (x−4)2
(notice the ± 'plus/minus' ..
there can be two square roots!)
there can be two square roots!)
So when we plot these two equations we should have a circle:
- y = 2 + √[25 − (x−4)2]
- y = 2 − √[25 − (x−4)2]
Waltr 2 6 25 X 4
Try plotting those functions on the Function Grapher.
It is also possible to use the Equation Grapher to do it all in one go.