Introduction to Ellipse equation

Introduction to Ellipse: 
An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed straight line is always constant and is always less than unity.
The constant ratio is generally denoted by e and is known as the eccentricity of the ellipse.
If S is the focus, ZZ’ is the directrix and P is any point on the ellipse, then by definition SP/PM = e => SP = e. PM.

General Equation of an Ellipse: x^2/a^2 + y^2/b^2 = 1. This is an Ellipse Equation.
Let us understand how to find Equation of Ellipse:
To understand how to Find the Equation of an Ellipse We have, x^2/a^2 + y^2/b^2 = 1, where a > b. ……..(i)So, y  = ± b/a sqrt(a^2 – x^2) ……..(ii) and x = ± a/b sqrt(b^2 – y^2) …….(iii).
We observe the following:
(a) Symmetry: For every value of x an equal and opposite values of y is found.
Similarly for every value of y an equal and opposite values of x is found as well. Thus the curve is symmetric about both the axes.
(b) Origin: The curve doesn’t pass from origin.
(c) Axes Intersection: The curve meets the x axis at y = 0. Putting y = 0 in (iii), we get x = ± a.
So the curve meets x-axis at A (a, 0) and A’ (-a, 0).
Putting x = 0 in (ii), we get y = ± b. So the curve meets y-axis at B(0, b) and B’ (0, -b). (d) Region: If x > a or x < -a, from (ii) we get imaginary values of y.
Therefore there is no part of the curve to the right of A or to the left of A’. If y > b or y < -b, from (iii) we get imaginary values of x.
Therefore there is no part of the curve above B(0, b) or below B’(0, -b). From (ii) we find that at x = 0, y = ± b and as x increases the value of y decreases and y = 0 at x = a.
Therefore the curve is a closed curve. With the help of the above facts and by joining some convenient points on the ellipse the general shape of the ellipse is x^2/a^2 + y^2/b^2 = 1.
Equations of ellipses in other forms: In the equation of the ellipse x^2/a^2 + y^2/b^2 = 1, if a > b or a^2 > b^2, then the major and minor axes lie along x-axis and y-axis respectively.
But if a < b or a^2 < b^2 (denominator of x^2 is less than that of y^2), then the major axis of the ellipse lies along the y-axis and is of length 2b and the minor axis along the x-axis and is of length 2a.
The coordinates of foci S and S’ are (0, be) and (0, -be) respectively. The equations of the directrices ZK and Z’K’ are y = ± b/e and eccentricity e is given by the formula a^2 = b^2 (1 – e^2) or e = sqrt(1 – a^2/b^2).

Ellipse Equation: If the centre of the ellipse is at the point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is (x – h)^2/a^2 + (y – k)^2/b^2 = 1. If we shift the origin at (h, k) without rotating the coordinate axes, then x = X + h and y = Y + k. So, the equation of ellipses in reference to new origin, becomes X^2/a^2 + Y^2/b^2 = 1.

Complementary angles

If a fifth grader asks you, what are complementary angles, you need not stare at him perplexed. This short note would give you enough information to answer that question.
Complementary angles definition is not as complicated as the name sounds. Take a look at the following figure:

We have two angles here. One is 40 degrees and the other is 50 degrees. But that is not important. The important point to be noted here is that 40 + 50 = 90 degrees. That means that the measure of these two angles add up to 90 degrees. With this understanding let us now write the definition of complementary-angles.

Definition: Complementary angles:

Two angles, whose sum is 90 degrees are said to be complementary-angles. The angles may or may not be adjacent angles. For example, see the figure below:

In this figure the two angle measures are 27 degrees and 63 degrees. Sum of these measures is 27 + 63 = 90 degrees. Therefore we can say that these two are a pair of complementary-angles. Examples of other such pair could be: 30 – 60, 45 – 45, 15 – 75, etc. There can be infinite such pairs. When a pair of angles are complementary-angles, we say that one is the compliment of the other, or one compliments the other. A pair of complimentary angles is always acute. That is to say that both the angles have to be acute angles.

Where do complementary angles occur?

Complementary angles are most common in a right triangle. In a right triangle since one angle is 90 degrees, the sum of the other two angles has to be 90 degrees. That is because we know that sum of all the three angles in a triangle should always be 180 degrees. Therefore in a right triangle, the two smaller angles are compliments of each other, or in other words they are complementary-angles.

The concept of complementary angles also finds its application in trigonometry. The sine ratio of an angle is always equal to the cosine ratio of the compliment of that angle. Similarly the tangent ratio of an angle is always equal to the cotangent ratio of the compliment of that angle. Also the secant ratio of an angle would be equal to the cosecant ratio of the compliment of that angle.

Standard deviations from the mean

Definition of standard deviation from the mean:
In Mathematics or Statistics, the standard deviation is a measure of dispersion from the mean. It tells us as to how far is the data spread from the mean value.

Calculating standard deviation of the mean:
The following steps are to be followed to calculate the standard deviation for a set of data {x1, x2, x3, x4….Xn} with mean = Xm.

Step 1: Make a table as follows:

Xi

X1

X2

X3

:

:

Xn

Step 2: Add a column to the table for Xi – Xm

Xi	Xi – Xm
X1	X1 – Xm
X2	X2 – Xm
X3	X3 – Xm
:
:
:
Xn	Xn – Xm

Step 3: Next square the difference thus found.

Xi	Xi – Xm	(Xi – Xm)^2
X1	X1 – Xm	(X1 – Xm)^2
X2	X2 – Xm	(X2 – Xm)^2
X3	X3 – Xm	(X3 – Xm)^2
:	:	:
:	:	:
:	:	:
Xn	Xn - Xm	(Xn – Xm)^2

Step 4: Find the sum of the values of the third column
= ∑(Xi – Xm)^2

Step 5: Divide the value obtained in step 4 by number of observations n
= [∑(Xi – Xm)^2]/n

Step 6: Standard deviation = SD = square root of the value found in step 5
Therefore,
SD = √{[∑(Xi – Xm)^2]/n}

Significance of the standard deviation from mean:
A measure of dispersion is a number showing how scattered the observations are from the mean. We already established above that standard deviation is a measure of dispersion.
If the difference between the observations and their mean are small, that is, if the observations are not too far away scattered from the mean, then the value of the standard deviation number is small. Such a data can be regarded as consistent or stable. In such cases, the mean can be considered as a good representative of the observations.

However, if the differences between the observations and their mean are large, that is, if the observations scattered far away from the mean, then the value of standard deviation would be large. Such a data cannot be regarded as stable and in some cases it may also be inconsistent. For such a data, the mean cannot be regarded as a good representative of the observations.
For example, consider the marks of two students X and Y in five subjects, as follows:

Subject	Student X	Student Y
A	58	86
B	52	39
C	45	12
D	48	68
E	57	55
Mean	52	52
SD	5.612	28.24

The mean marks of both the students are the same but since student X has a lower standard deviation, we can say that he is a more consistent performer.

Quartile: As a measure of central tendency

In statistics, just like, mean, median, mode etc, it is also a measure of central tendency. It is also used to measure dispersion. As the name suggests, divides the data into four quarters. There are many types of these and others are deciles, percentile etc.

Types of quartiles:
There are three types of these and they are as follows:
1. Lower quartile
2. Middle quartile
3. Upper quartile

Lower quartile definition: The first one fourth of the range is called the first quarter. The two end points of this first quarter are the lowest value and the lower quarter. The lower quarter is also called the first quarter. It is denoted by “Q1”. The lower quarter also corresponds to the 25th percentile. It can be calculated using the formula:
L = n*(1/4). Where L = the position of the quarter value. That means that starting from the first number of the data the Lth number would be the first quarter. N = total number of data points.

Middle quarter definition: The middle quarter is also called the median. It divides the data into exactly two halves. Median is a popular measure of central tendency. If there are total n number of observations, then the (n+1)/2 th observation would be the median. Median is denoted by “Q2” or M. The middle quarter corresponds to the 50th percentile.

Upper quartile definition: The upper-quarter divided the range into two parts in the ration 1:3. That means the first 3/4th of the values fall below the upper-quarter and the last 1/4th of the values fall above the upper-quarter. The upper-quarter is also known as the third quarter. It corresponds to 75 percentile. It is denoted by “Q3”.  The formula for calculating the upper-quarter is as follows:
N = n*(3/4). Here, n = number of observations. The Nth observation would be the upper-quarter.

Example: Find the lower, middle and upper-quarter of the following data: 50, 80, 40, 40, 60, 30, 80.
Solution: First we arrange the data in ascending order from the lowest to the highest.
30, 40, 40, 50, 60, 80, 80
Next we locate the middle number. Which, in our example, is 50. So the median = 50.
Now for the lower quarter, we consider the first half of the data only and find the middle number of that. So that would be 40. Lastly the upper quarter would be 80.

Best Measure of Central Tendency

What are Measures of Central Tendency?
Central tendency refers to the centre value of the data. Measures of the Central Tendency calculate the central or middle value or the average of the given data. This is different from the measures of dispersion, which tells about the spread of the data values around the average.  We can thus define Measures of Central Tendency as the methods of measuring the central tendency of the given data. There are various measures of the central tendency. The measures of the central tendency are mean, median and mode. Each of these measures differs in the way they calculate the central tendency or average of the given data.

The method of calculating mathematical average is termed as arithmetic mean. The measure that finds out the value which is present half-way in the data arranged in ascending order is termed as median. Most frequently occurring data is called as mode.

Features of Measures of Central Tendency
• For a normal data set, all the three measures give equal results.
• Unlike mode, the mean and median have only one value as the result.

Best Measure of Central Tendency
The best measure of central tendency depends upon the given data set, situation and the nature of analysis for which the central tendency has to be calculated. Let us see different situations and the appropriate measure of central tendency that is best suited for those situations.

Mean, Median or Mode
Any one of the three measures of central tendency mean, median or mode can be used to calculate the central tendency in case below:
• Data set containing numerical data or interval data.
• Data set containing continuous data.

Mean
It is the commonly used measure of central tendency for:
• Data set having mathematical data values.
• Data set used for statistical calculations or interference.
• Data set having no extreme values.
• Data set having symmetrical values.

Median
The data sets whose central tendency is to be measured using median are:
• Data set containing random numbers.
• Data set containing different numbers.
• Data set containing extreme values.
• Data set containing ordinal data or ranked data.

Mode
The data set whose central tendency is to be measured using mode are:
• Data set containing values that are closer or equal in value.
• Data set containing nominal data. Mode is the only measure of central tendency to measure nominal data or qualitative data.   For example, Marital status

There are two scenarios when either median or mode can be applied. They are:
• Data set containing outliers
• Data set where data is skewed

Introduction to Basic Algebra For Kids

Algebra is the main part of mathematics in which symbols or signs such as letters of the alphabet which stands for the numbers. The following algebra based on the kids only.

There are two opposite sets of numbers: one is positive numbers and another is negative numbers.

Positive Numbers:

All positive integer numbers are the positive numbers. It is including the plus sign. The numbers are called co efficient and x is the variable.

Example: x, 2x, 3x, 4x, 5x…

Negative Numbers:

The Negative numbers must go to the left and the positive to the right. While the negative numbers are always written in a minus sign in front of them, the positive numbers do not represent always sign in front of them.

Example: -x,-2x,-3x …

Basic Algebra for Kids Addition and Subtraction:

Adding and Subtracting of two numbers:

The Addition basic algebra is the adding the positive numbers or negative numbers. If the both numbers are positive numbers or negative numbers adding the algebra expression.

Basic Algebra kids example for addition:

3x + 4x = -7y –y

7x = -6y

7x + 6y = 0

The subtraction of basic algebra for kids it is the subtracting the positive numbers to negative numbers or negative numbers to positive numbers.

Basic Algebra kids example for subtraction:

4x-3x = 6y – 7y

x = -y

x + y=0

Multiplication and Division of Basic Algebra Numbers for Kids:

The multiplication of positive and negative numbers has some rules which depend on the equation. It is also called simplifying.

Example for Basic Algebra kids multiplication:

(2x + 3) (3x + 2)

(2x)(3x + 2) + (3) (3x + 2)

6x2+4x = 9x +6

6x2 +4x -9x -6 = 0

6x2 – 5x -6 =0

Dividing of two numbers:

Dividing positive and negative numbers has certain special rules. The division is containing divisor, quotient, and remainder.

Example for Basic Algebra kids Division:

(4x + 6) / (2x+3)

2x+3) 4x + 6 (2
4x + 6
-----------
0
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Introduction of slope in math

In mathematics, the slope or incline of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is

m=`(y2-y1)/ (x2-x1)`

(Source: Wikipedia)

Definition of Constant Slope in Math:

In math, to finding constant slope to build up a slope formula. To begin, we choose points P and Q on the line. To distinguish between the corresponding of these points, we use subscript notation. Point P has coordinates (x1,y1) and point Q has coordinated(x2.y2).

As we move from point P to point Q, the rise is the dissimilarity of the x-coordinates:x2-x1, as the constant slope is the ratio `(rise) / (run)` ,  we have the following formula for calculating constant slope.

In math, another notation that we use to describe constant slope involves the symbol ∆, which is the letter delta from the Greek alphabet. If the modify in y is represented by ∆y and the change in x is represented by ∆x, then:

M=`(Deltay) /(Deltax)` where ∆x≠0

The entire notation associated with the concept of constant slope of a line.

Example for Constant Slope in Math:

In math, find the constant slope of the line passing through (-1,2) and (3,-4)

Strategy we will use the constant slope formula to find the constant slope.

Why?  We  know the coordinates of two points on the line.

Solution:

We can let(x1,y1)=(-1,2) and (x2,y2)=(3,-4). Then we have

m=`(y2-y1)/(x2-x1)` this is the constant slope formula.

=`(-4-2)/(3-(-1))`      substitute -4 for y2, 2 for y1,3 for x2, and -1 for x1.

=`(-6/4)`

=`-3/2`

Write `(-3/2)` with the – sign in front of the fraction. The outcome is negative.

The constant slope of the line is `-3/2`

In math, the line passing through (-1,2) and (3,-4) points. Notice that we obtain the same result when the constant slope of the line is

m=`(rise)/(run)`=`-3/2`