Externally Tangent Circles

Two intersect circles in a single point is known as tangent circles. It can be divided into two types of tangency: internal and external. By using the tangent circles many problems and constructions in geometry are solved. This type of problems have real-life applications. The followings are some real-time applications: trilateration and maximizing the use of materials.

                                             

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Illustration:

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called Common External Tangent. A common tangent can be any one of the following:External tangent or Internal tangent.

                                                   
          From the above figure, we can see that line PQ is the common external tangent.

                                               
          From the above figure, we can see that line AB is the common external tangent.

Theorem for Externally Tangent Circles:

Theorem:If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
                                                       
Solution:
            If:        Line  AB is a tangent
                       D is point of tangent.
            Then:   OD ┴ AB

Example problems:

Example 1: Find the common external tangent from the below figure.

                                             
Solution:
            A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called common external tangent.
            From the given figure, the line CD touches both the circles on the same side of the line is the external tangent to the circle.
            Therefore, CD is the common external tangent line.

Example 2: Find the common external tangent from the below figure.
                                                                                    
Solution:
            The given figure is concentric circles. Concentric circles having the same center and not tangent.

Example 3: Find the common external tangent from the below figure.
                                                     
Solution:
          From the above figure, we can identify that given is two complete separate circles.
          The given figure have two external tangents.

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Example 4: Find the equation for tangent lines y=3x³ from the point x= 2.
Solution :
Given, y=3x³………….(1)
Step 1: To find the first derivative of y

                y ' = 3x ² ………..(2)
Step  2: To plug x =2 into y ' to find the slope at x.
                y ' = 3(2) ²
                y= 12
               Slope of tangent line =12
Step 3:To plug x=2 into y to find the y value of the tangent point
              y =3(2)³
              y=24 
              Hence tangent point (2,24)
Step 4: To plug the slope=12 and point (2,24) using the point-slope equation to find the equation     for the tangent line.
              y- y₁ =m (x-x₁)
              y- 24 =12 (x-2)
              y =12x- 24 + 24
              y=12x
Example 5: Find the slope of tangent line which is passing through the point (2, 0) and the curve y = 2x – x³
Solution:
            Step 1: Find the derivative y¹
                                y = 2x – x³
                                y¹ = 2 -3x²
            Step 2: To find the slope of tangent line,
                         Given point (2, 0) --- > (x, y)
                         y¹ = 2 -3(2)² ---- > 2 – 3(4) --- > 2 -12 = -10
                         Slope of the tangent line = -10         

Practice problems:

1. Determine the equation of the tangent line passing through the point (− 1, − 2) and having slope4 /7
Answer:   4x − 7y − 10 = 0
2. Determine the equation of the line with the slope 3 and y-intercept 4.
Answer: y = 3x + 4

Power Factor Conversion

To signify a lot of the numbers we have used scientific notation. For example the number 7500 is identified as 7.5E3. Here the letter “E" can be read as "times ten to the power of". Every unit is written out in full before its contraction is used. Some units have superscripts in them. In power factor conversion includes the watt to joule, foot pound to force per second, kilogram to meter per second, horse power to watt.

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Example problems of power factor conversion:

Power factor conversion Problem 1:

Convert watt to joule per second.

25 W

30 W

Solution:

We can convert watt to joules per second by using the following method.

1 watt = 1 joule per second

The initial unit is 25 W and 30 W

Here conversion factor is 1 W/ 1 joule/sec

That is, `25*` `1/1` = 25

`30*` `1/1` =30

Therefore the final unit is 25 and 30 units

Answer: 25 joule/sec and 30 joule/sec

Power factor conversion Problem 2:

Convert watt to joule per minute.

55

Solution:

We can convert watt to joule per minute by using the following method.

1 watt = 60 joule per minute

The initial unit is 55 W

Here conversion factor is 1 watt = 60 joule per minute.

That is, `55*` `60/1` =3300

Therefore the final unit is 3300 units

Answer: 3300 joule/minute

Power factor conversion Problem 3:

Convert watt to horsepower.

25

Solution:

We can convert watt to horsepower by using the following method.

1 watt = 0.001341022 horsepower

The initial unit is 25 watt

Here conversion factor is `(0.001341022 hp)/(1 watt)`

That is, 25*0.001341022= 0.033525552 horsepower

Therefore the final unit is 0.033525552 horsepower

Answer: 0.033525552 horsepower

Practice problems of power factor conversion:

1. Convert watt to joule per second: 75

2. Convert watt to joule per minute: 22

3. Convert watt to horsepower: 45

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Answer:

75 joule per second
1320 joule per minute

3.  0.060345994 horsepower

Prime and Rate

Prime: Definition of prime number is a number which cannot divide by any other numbers. That is ,only the number itself and 1 can divide the prime number.
        Rate: Definition of rate is the ratio between the distance and time. Definition of rate is always in the form of fraction.
Let us see the definition of prime and rate.

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Definition of prime number:

Definition of prime number:

If a number is not allowed to divide by any number then it is called as prime number. Other wise it is called as composite number.

Example: 133

This number can not be divide by any number. Only 1 and 133 can divide it. So, 133 is considered as prime number.

Example:1

Find the prime number among the following numbers.

a. 155

b. 242

c. 141

d. 339

Ans :141

Example: 2

Find the prime number among the following numbers.

 a.122

b.223

c.144

d.166

Ans : 223

Definition of rate:

Definition of rate:
  Formula for finding a rate is,

Rate=Distance/Time

        Rate has a units depends on the units of distance and time. If the distance has a unit in km and the time is in hr then the rate should be written as km/hr.
Example problems for finding a rate:
Example:1

Find the rate of scooter. The distance travelled by a scooter is 89 km and the time taken for that is 3 hours.

Solution:
Given,

Distance = 89km

Time=3 hr

We know the rate formula is Rate=distance/time

Rate=`89/3`km/hr.

Therefore the rate of the given scooter is `89/3` km/hr.

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Example:2

Find the time taken by a cycle. The distance travelled by a cycle is 24 km and the rate is `12/6`km/hr.

Solution:
Given,

Distance = 24km

Rate=`12/6`km/hr

We know the rate formula is Rate=distance/time

Time= Distance/time

         =`24/(12/6)`

        =`144/12`

       =12 hours.

Therefore the time taken by a cycle to travel 24 km is 12 hours.

Fraction Conversion Table

Fraction:
A fraction is a number that can represent part of a whole. The earliest fractions are reciprocals of integers: ancient symbols represent one part of two, one part of three, one part of four, and so on. A much later development was the common fractions which are used today (½, ⅝, ¾, etc.) and which consist of a numerator and a denominator.

Fraction can be converted into to forms as shown below,

• Fraction conversion into decimal
• Fraction conversion into percentage.

Through this article you can learn about fraction conversion table, how to make conversion of fraction into decimal and how to make conversion of fraction into percentage.

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Example for fraction conversion table:

Fraction conversion table:

         We can express the conversion of fraction in terms of table format.This table is known as fraction conversion table.The following are the two examples for fraction conversion table.

                       

Fig(i) Fraction conversion table 1                                      Fig(ii) Fraction conversion table 2

The fractions that contains the denominator as a  power of  10 called as decimal fractions.
Ex:
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1000

Problems on fraction conversion table:

Pro 1: Convert the fraction 45 / 90
Sol:
Given, 45 / 90 into decimal.
To find the decimal,divide the numerator by the denominator ,
             _________
         90 )  45      (
45 cannot be divided by 90.
So Multiply 45 by 10 and add one decimal place in quotient,
             _________
         90 )  450     ( 0.5
                450     
                 0        
So the decimal of fraction 45/90 is 0.5
Ans: 45/90 = 0.5
Pro 2: Convert the fraction 76 / 43 into decimal.
Sol:
Given, 76 / 43 as decimal
To find the decimal,divide the numerator by the denominator ,
             _________
         43)  76     ( 1.76
               43     
                33      
33 cannot be divide by 43. So multiply 33 by 10 and one decimal place in quotient,
            _________
         43)  76     ( 1.7
               43     
               330    
               301   
                 290
                 258
                   38
The decimal for 76 / 43 = 1.76
Ans:76 / 43 = 1.76

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Pro 3: Convert the fraction 6 / 5 into decimal.
Sol:
Given, 6 / 5 as decimal
To find the decimal,divide the numerator by the denominator ,
           _________
        5 )  6     ( 1
              5     
              1   
1 cannot be divide by 5. So multiply 1 by 10 and one decimal place in quotient,
            _________
           5 )  6     ( 1.2
                 5     
                 10
                 10  
                   0 
The decimal for 6/5 = 1.2
Ans: 6/5 = 1.2

Venn Diagram Example of Disjoint

Let use see about venn diagram example of disjoint.Venn diagrams also called as the set diagram.These are diagrams that show all theoretically possible logical relations between finite groups of set. Venn diagrams were considered approximately 1880 by John Venn. They are used to teach elementary set theory, and show simple set relationships in logic, probability and statistics.


venn diagram example of disjoint - Notations


Curly  braces - {...} - are used to phrase.
These braces can be used in various ways.
For example:

• List the elements of a set: {-3, -2, -1, 0, 1, 2, 3,4}

• Describe the elements of a set: {integers between -3 and 3 inclusive}

• Use an identifier (the letter x for example) to symbolize a typical element, a '|' symbol to stand for the axiom such that', and then the rule or rules that the identifier must follow: {x | x is an integer and |x| < 5}

The Greek letter ∈ is used as follows:

• ∈ means 'is an element of ...'. For example: 3 ∈ {positive integer}

• ∉ means 'is not an element of ...'. For example: Washington DC ∉ {European capital cities}

• The set is a finite: {British citizens}

                      Or

•  infinite: {6, 12, 21, 24, 35, ...}

Sets are usually be represented using upper case letters: A, B,X,Z ...

venn diagram example of disjoint - Example

The following is the diagram representation of disjoint in venn diagram.
Two sets are equally exclusive also called disjoint. If do not have any elements in common and need not together contain the universal set.
The following venn diagram represents the disjoint sets.

Set of all elements of A is also known as difference of set A-B, which do not go to B. In the set planner form, the difference set is :   

 

Example problem for disjoint set.
A={2,3,4,1,8,9}   B={2,3,4,1,10,12} What is the A-B and B-A?
Solution:
A-B=?
Given A={2,3,4,1,8,9}
            B={2,3,4,1,9,10,12}
Here all elements of A  an available in B except 9.
So the A-B is 9.
B-A=?
Here all elements of A  an available in A except 12.
So the B-A is 12.

Line Intercept Sampling

In this article, we will discuss about the line intercept sampling. Sampling means method of selecting sample. The line intercepts have two types, one is x intercept and next one is y intercept. X intercept means that, the point crosses the x -coordinates or axis and y intercept means point crosses the y- axis of the line. The slope intercepts form of line y = mx + b, where m is slope of the line, and b - the y intercept. Let us learn about the line intercept sampling example problems are given below.

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Example problems for line intercept sampling:

Example problem 1:

Find the x intercept of the line, 11.5x + 13.5y = 27

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the x- intercept Plug y = 0 in the equation

Here x intercept, so y = 0

11.5x + 13.5(0) = 27

11.5x = 27

Divide by 11.5 on both sides.

` (11.5x)/(11.5)` = ` (27)/(11.5)`

After simplify this, we get

x = 2.34

x intercept = (2.34, 0)

Example problem 2:

Find the y intercept of the line, 4x + 24y = 72

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

4(0) + 24y = 72

0 + 24y = 72

24y = 72

Divide by 24 on both sides.

`(24y)/(24)` = ` (72)/(24)`

After simplify this, we get

y = 3

y intercept = (0, 3).

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More example problems for line intercept sampling:

Example problem 3:

Find the y intercept of the line, 5x + 30y = 120

Solution:

The slope intercept form of line y = mx + b, where m is the slope of the line

Given equation is in the form of ax + by = c. To find the y- intercept Plug x = 0 in the equation

Here y intercept, so x = 0

5(0) + 30y = 120

0 + 30y = 120

30y = 120

Divide by 30 on both sides.

`(30y)/(30)` = ` (120)/(30)`

After simplify this, we get

y = 4

y intercept = (0, 4)

The above examples are helpful to learn of line intercept sampling.

Alternating Current Generator

The household electricity that we used is based on the principle of Alternating Current. One advantage of Alternating Current is that it does not induce fatal shock. Also it has an added advantage of easy voltage amplification. In this article we shall discuss the Alternating Current Generator which is often abbreviated as A.C. Generator.

Understanding Alternating Exterior Angles is always challenging for me but thanks to all math help websites to help me out. 

Introduction to A.C. generator

The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating current (A.C.).

Principle of ac generator

One method to induce an emf or current in a loop is through a change in the loop orientation or a change in it’s effective area. As the coil rotates in a magnetic field B , the effective area of the loop (the face perpendicular to the field) is AcosӨ, where Ө is the angle between A and B. This method of producing a flux change is the principle of operation of a simple ac generator.

A. C. Generator : Construction and Working

It consists of a coil mounted on a rotor shaft. The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The coil called the armature is mechanically rotated in the uniform magnetic field by some external means . The rotation  of the coil causes the magnetic flux through it to change, so an emf is induced in the coil . The ends of the coil are connected to an external circuit by means of slip rings and brushes.

Working
When the coil is rotated with the constant angular speed of w , the angle Ө between the magnetic field vector B and the area vector A of the coil at any instant t is Ө=wt(assuming Ө=0⁰at t=0). As a result, the effective area of the coil exposed to the magnetic field lines changes with time, and the flux at any time is ф_B=BAcosӨ=BAcoswt From faraday’s law, the induced emf for the rotating coil of N turns is then,  E=-N×dф_B/dt = -NBA×dcoswt∕dt thus, the instantaneous value of the emf is  E= NBAwsinwt.Where NBAw is the maximum value of the emf,which occurs when sinwt= +1 or -1. If we denote NBAw as E₀, then E=E₀sinwt since the value of the sine function varies between +1 or -1, the sign, or polarity of the emf changes with time. 

My forthcoming post is on Perfect Negative Correlation and nmat 2013 syllabus will give you more understanding about Algebra.

The emf has its extremum value when Ө=90⁰ or Ө=270, as the change of flux is greatest at these points. The direction current changes periodically and therefore the current is called alternating current.
The modern ac generator with a typical output capacity of 100 MW is a highly evolved machine.