Conic Sections

Introduction:

Algebra is the one of the most important chapter in mathematics subject. Here the Conic section is one of the topics in algebra chapter. In mathematics a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. The conic sections were named and studied as long ago as 200 BC.

Conic Sections:

Conic section is generally defined as intersection of a plane and a cone, it will  depend on How the plane is oriented, the curve will be one conic sections. Or a conic section is a curve formed by the intersection of a cone with a plane.
 Four types of conic sections are there,

• Circle,
• Parabola,
• Ellipse, and
• Hyperbola.

Conics: Ellipses: Introduction

Conics: Ellipses: Introduction
An ellipse, informally, is an oval or a "squished" circle. In "primitive" geometrical terms, an ellipse is the figure you can draw in the sand by the following process: Push two sticks into the sand. Take a piece of string and form a loop that is big enough to go around the two sticks and still have some slack. Take a third stick, hook it inside the string loop, pull the loop taut by pulling the stick away from the first two sticks, and drag that third stick through the sand at the furthest distance the loop will allow. The resulting shape drawn in the sand is an ellipse.
Each of the two sticks you first pushed into the sand is a "focus" of the ellipse; the two together are called "foci" (FOH-siy). If you draw a line in the sand "through" these two sticks, from one end of the ellipse to the other, this will mark the "major" axis of the ellipse. The points where the major axis touches the ellipse are the "vertices" of the ellipse. The point midway between the two sticks is the "center" of the ellipse.

Equation of Ellipse with Standard Positions

Equation	X2/a2+y2/b2=1, a > b	X2/a2+y2/b2=1, a <>
Centre	(0,0)	(0,0)
Coordinate of centre.	(a,0),(-a,0)	(0,b),(0,-b)
Length major axis	2a	2b
Length of minor axis	2b	2a
Equation of major axis	Y = 0	X = 0
Equation of minor axis	X = 0	Y = 0
Equation of directrix	X = ± a/e	Y = ± b/e
Eccentricity Latus rectum	‘e’ = √(1-b2/a2) 2b2/a	‘e’ = √(1-a2/b2) 2a2/b
Focal distance	a ± ex	b ± ey
Coordinates of foci.	(ae , 0), (- ae , 0)	(0 , be). (0 , - be)

Hope you like the above Equation of Ellipse with Standard Positions Expression
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Translating Word Problems

Translating Word Problems: The first step to effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and what you still need.

The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknows, clearly labelling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:
The third step is to look for "key" words. Certain words indicate certain mathematical operations. Below is a partial list.
Addition- increased by more than combined, together total of sum added to
Subtraction- decreased by minus, less difference between/of less than, fewer than
Multiplication- of times, multiplied by product of increased/decreased by a factor of (this type can involve both addition or subtraction and multiplication!)
Division- per, a out of ratio of, quotient of percent (divide by 100)
Equals- is, are, was, were, will be gives, yields sold for

Translating Word Problems: Examples

• Translate "the sum of 8 and y" into an algebraic expression.
  This translates to "8 + y"

• Translate "4 less than x" into an algebraic expression.
  This translates to "x – 4"
  
  Remember? "Less than" is backwards in the math from how you say it in words!

•  Translate "x multiplied by 13" into an algebraic expression.
  This translates to "13x"

• Translate "the quotient of x and 3" into an algebraic expression.
  This translates to " x/3"

• Translate "the difference of 5 and y" into an algebraic expression.
  This translates to "5 – y"

• Translate "the ratio of 9 more than x to x" into an algebraic expression.
  This translates to "(x + 9) / x"

• Translate "nine less than the total of a number and two" into an algebraic expression, and simplify.
  This translates to "(n + 2) – 9", which then simplifies to "n – 7"
  
  Here are some more wordy examples:

• The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.
  
  Whatever the width w is, the length is 30 more than this. Recall that "more than" means "plus that much", so you'll be adding 30 to w.

         The expression they're looking for is "w + 30".

Hope you like the above example of Translating word Problems
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Translating words into Algebraic Expression

Definition: Translating words into algebraic expression is the process of translating the word problems into an algebraic expression which can be used to solve the word problem and produce the solution for the given words problem.

Problems of translating words into algebraic expression:

Problem 1:

Flowers shop has thirty Roses and forty Lilly. How many pieces of flowers does flowers shop have?

Solution:

Let a = Total number of Flowers in the flowers shop.
The sum of thirty Roses and forty Lilly is equal to the total number of flowers in the flowers shop. It translates the words problem into an algebraic expression.
a = 30 + 40

Solve this expression.

Let a = Total number Pieces of Flowers in the flowers shop
a = 70.

There are 70 Pieces of Flowers in the flowers shop.

Problem 2:

The sum of twice a number plus 20 is 86.Find the number.

Solution:

In words problem, the word is means equals and the word and means plus.
Translating words problem into an algebraic expression
the sum of twice a number and 20 equals 86.
Write an expression

2X + 20 = 86

Solve this expression using the variable.

2X + 20 = 86 (Expression)

2X + 20 – 20 = 86 - 20 (subtract by -20 on both sides)

2X= 66

X = 33 (Divided by 2 on both sides we get the result)

Solution to the problem is 23.

Hope you like the above example of Translating words into Algebraic Expression
Please leave your comments, if you have any doubts.

Ratio and Proportion

Ratio: Ratio is the numerical relationship between two quantities of the same kind. The first quantity is called the antecedent and the second quantity is called the consequent.

Proportion: a, b, c and d are said to be proportion if a : b = c : d.
a and d are called the extremes, b and c are called the means. a, b, c and d are called first proportion, second proportion, third proportion and fourth proportion respectively.

Types Of Ratios and Proportion:

Ratios are classified in to six types:

1) Duplicate ratio:The ratio of the squares of the two numbers.Ex: 9:16 is the duplicate ratio of 3:4.

2) Triplicate ratio:The ratio of the cubes of the two numbers.Ex: 27:64 is the triplicate ratio of 3:4.

3) Sub-duplicate ratio:The ratio between square root of the two numbers.Ex: 4:5 is the sub-duplicate ratio of 16:25.

4) Sub-triplicate ratio:The ratio between the cube roots of the two numbers.Ex: 4:5 is the sub-triplicate of 64:125.

5) Inverse ratio:If the two terms in the ratio interchange their places,then the new ratio is inverse ratio of the first.Ex: 9:5 is the inverse ratio of 5:9.

6) Compound ratio:The ratio of the product of the first term to that of the second term of two or more ratios.Ex: 3/4,5/7,4/5 and 3/5 is 3/4 x 5/7 x 4/5 x 3/5 = 9/35.

Proportions are classified in to four types:

1) Continued proportion:In the proportion 8/12=12/18, 8,12,18 are in the continued proportion.

2) Fourth proportion:If a:b=c:x,then x is called forth proportion of a,b and c.The fourth proportion of a,b.c =bc/a

3) Third proportion:If a:b=b:x,then x is called third proportion of a and b.Third proportion of a,b =b^2/a

4) Second or mean proportion: If a:x=x:b,then x is called second or mean proportion of a and b.Therefore mean proportion of a and b =Root of(ab)

Simple Interest and Compound Interest

Interest: Interest is the amount of money we pay for the use of some amount of money

There are two types of interests,

a) simple interest: Simple interest is the Interest paid / compensated only on the original principal, not on the interest accrued

b) compound interest: interest means that the interest Which includes the interest calculated on principal amount

Algebra Formula to find simple interest :

Simple interest I = PRT

Where, P is the Principal amount, R is the Rate of interest, N is Time duration.

When we knows interest I we can find p, n or r using the same formula ,

Different forms of algebra simple interest formula

Algebra formula to find Compound Interest:

FV = PV (1+r)ⁿ

PV is the present value

r is the annual rate of interest (percentage)

n is the number of years the amount is deposit or borrowed for.

FV = Future Value is the amount of money accumulate after n years, including interest.

Elementary Algebra, ratio word problems

In Elementary Algebra practice , a ratio expresses the magnitude of quantities relative to each other rather. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Mathematically, a proportion is defined as the equality of two ratios. However in common usage the word proportion is used to indicate a ratio, especially the ratio of a part to a whole.Let's see an example from 4th grade math algebra word problems with solution

Question:-

Find two numbers such that their difference ,sum and product are in the ratio 1:4:15 respectively.

Answer:-

This solution will explain how to understand ratio word problems

Let the numbers be x and y

By the problem

(x-y) : (x+y) : (xy) = 1 : 4 : 15

x-y = k ----- 1

x+y = 4k ----- 2

xy = 15k -------- 3

Where k is the constant of proportionality

From 1+2 , we get x = 5k/2

Substitute this in eq 2 , we get y = 3k/2

By substituting this x and y values in eq 3

We get k = 4

If we put this k value in x = 5k/2 , we get x = 10

If we put k value in y = 3k/2 , we get y = 6

So the numbers are 10,6