- Also, while 1:: 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1: 2: easier to remember.) The cited theorems are from the Appendix, Some theorems of plane geometry. Here are examples of how we take advantage of knowing those ratios. First, we can evaluate the functions of 60° and 30°.
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-->This topic explains how to install the Windows PowerShell 2.0 Engine.
Windows PowerShell 3.0 is designed to be backwards compatible with Windows PowerShell 2.0. Cmdlets,providers, snap-ins, modules, and scripts written for Windows PowerShell 2.0 run unchanged inWindows PowerShell 3.0 and Windows PowerShell 4.0. However, due to a change in the runtimeactivation policy in Microsoft .NET Framework 4, Windows PowerShell host programs that were writtenfor Windows PowerShell 2.0 and compiled with Common Language Runtime (CLR) 2.0 cannot run withoutmodification in later releases of Windows PowerShell, which is compiled with CLR 4.0.
To maintain backward compatibility with commands and host programs that are affected by thesechanges, the Windows PowerShell 2.0, Windows PowerShell 3.0, and Windows PowerShell 4.0 engines aredesigned to run side-by-side. Also, the Windows PowerShell 2.0 Engine is included in Windows Server2012 R2, Windows 8.1, Windows 8, Windows Server 2012, and Windows Management Framework 3.0. TheWindows PowerShell 2.0 Engine is intended to be used only when an existing script or host programcannot run because it is incompatible with Windows PowerShell 3.0, Windows PowerShell 4.0, orMicrosoft .NET Framework 4. Such cases are expected to be rare.
The Windows PowerShell 2.0 Engine is an optional feature of Windows Server 2012 R2, Windows 8.1,Windows® 8 and Windows Server® 2012. On earlier versions of Windows, when you install WindowsManagement Framework 3.0, the Windows PowerShell 3.0 installation completely replaces the WindowsPowerShell 2.0 installation in the Windows PowerShell installation directory. However, the WindowsPowerShell 2.0 Engine is retained.
For information about starting the Windows PowerShell 2.0 Engine, seeStarting the Windows PowerShell 2.0 Engine.
On Windows 8.1 and Windows 8
On Windows 8.1 and Windows 8, the Windows PowerShell 2.0 Engine feature is turned on by default.However, to use it, you need to turn on the option for Microsoft .NET Framework 3.5, which itrequires. This section also explains how to turn the Windows PowerShell 2.0 Engine feature on andoff.
To turn on .NET Framework 3.5
On the Start screen, type Windows Features.
On the Apps bar, click Settings, and then click Turn Windows features on or off.
In the Windows Features box, click .NET Framework 3.5 (includes .NET 2.0 and 3.0 toselect it.
When you select .NET Framework 3.5 (includes .NET 2.0 and 3.0, the box fills to indicate thatonly part of the feature is selected. However, this is sufficient for the Windows PowerShell 2.0Engine.
To turn the Windows PowerShell 2.0 Engine on and off
On the Start screen, type Windows Features.
On the Apps bar, click Settings, and then click Turn Windows features on or off.
In the Windows Features box, expand the Windows PowerShell 2.0 node, and click theWindows PowerShell 2.0 Engine box to select or clear it.
On Windows Server 2012 R2 and Windows Server 2012
Use the following procedures to add the Windows PowerShell 2.0 Engine and Microsoft .NET Framework3.5 features. The Windows PowerShell 2.0 Engine requires Microsoft .NET Framework 2.0.50727 at aminimum. This requirement is fulfilled by Microsoft .NET Framework 3.5.
To add the .NET Framework 3.5 feature
In Server Manager, from the Manage menu, select Add Roles and Features.
Or in Server Manager, click All Servers, right-click a server name, and then selectAdd Roles and Features.
On the Installation Type page, select Role-based or feature-based installation.
On the Features page, expand the .NET 3.5 Framework Features node and select .NETFramework 3.5 (includes .NET 2.0 and 3.0).
The other options under that node are not required for the Windows PowerShell 2.0 Engine.
To add the Windows PowerShell 2.0 Engine feature
In Server Manager, from the Manage menu, select Add Roles and Features.
Or Server Manager, click All Servers, right-click a server name, and then select AddRoles and Features.
On the Installation Type page, select Role-based or feature-based installation.
On the Features page, expand the Windows PowerShell (Installed) node and select WindowsPowerShell 2.0 Engine.
For information about starting the Windows PowerShell 2.0 Engine, seeStarting the Windows PowerShell 2.0 Engine.
On Earlier Systems
The Windows Management Framework 4.0 package thatinstalls Windows PowerShell 4.0 on Windows 7, Windows Server 2008 R2, and Windows Server 2012,includes the Windows PowerShell 2.0 Engine. The Windows PowerShell 2.0 Engine is enabled and readyto use, if necessary, without additional installation, setup, or configuration.
The Windows Management Framework 3.0 package that installs Windows PowerShell 3.0 on Windows 7,Windows Server 2008 R2, and Windows Server 2008, includes the Windows PowerShell 2.0 Engine. TheWindows PowerShell 2.0 Engine is enabled and ready to use, if necessary, without additionalinstallation, setup, or configuration.
See Also
A circle is easy to make:
Draw a curve that is 'radius' away
from a central point.
And so:
All points are the same distance
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
(The ± means there are two possible values: one with + the other with −)
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).
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:
(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
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
Hype 3 3 5 5. 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
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:
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
Sidewriter 1 2 0 2
Rearrange to get 'y=':
there can be two square roots!)
So when we plot these two equations we should have a circle:
Sidewriter 1 2 0 Mm
- y = 2 + √[25 − (x−4)2]
- y = 2 − √[25 − (x−4)2]
Sidewriter 1 2 0 1
Try plotting those functions on the Function Grapher.
It is also possible to use the Equation Grapher to do it all in one go.
