Equation of the line which is passing through two points

Question :-

Find the equation of the line which is passing through two points (-3,7)(5,-1)

Answer:-

We have to use the point formula to find the equation of the line which is similar to midpoint formula

y-y1     x-x1
------ = ------
y2-y1     x2-x1

We have 2 points

( -3 , 7 )  and ( 5 , -1 )
  x1  y1          x2  y2

So the equation is

y-7      x-(-3)
------ = ------
-1-7      5-(-3)

y-7      x+3
------ = ------
-8        8

We can further simplify it by cross multiplication.that comes under indices maths

similarly we can find all points having an x-coordinate of 2 whose distance from the point 2 1 is 5

Simple quadratic equation

Topic: Quadratic Equation


An equation of the form ax²+bx+c=0 where a, b, c are real numbers and a =/ 0, is called a quadratic equation.

Question:

Solve: x2 - y = 3, x - y = -3

Answer:

   x2 - y = 3, x - y = -3

   x2 - y = 3

    x  - y = -3

    + y + y
    __________
    x = y - 3

  Putting this in the first equation

  (y -3)x2 - y = 3

 y2 - 6x + 9 - = 3

 y2 - 6x + 9 - y = 3

 y2 - 7y + 9 = 3

        -3    -3
        ____________

Factoring (y - 6) (y -1) = 0

         y = 6 on 1

         x = y - 3 = 6 - 3 = 3

         x = 1 - 3 = - 2

  The solution set is : {(-2,1) , (3,6)}

Graphing trigonometric functions

Graphs are useful for analyzing properties of various trigonometric functions and are valuable in many applications.The most common use of these functions is in analyzing waves ,sound and electric current and voltage.Although any trig function can be graphed,the emphasis here is on graphing trigonometric functions.

Graphing a Sine function

The graph of y= sinx(where x equals the angle) can be sketched by simply constructing a table ,like the one shown below,selecting values for the angle,x and solve for y,then plotting the points on a graph .

x(2)    : -90   45    0    45      90   180   270   360 

y(sin2) : -1   -0.8   0   0.707     1     0    -1     0

The result graph below,which continues indefinitely in both directions

Note:- The x-axis is set in increments of standard angles in degrees,although radians can also be used .
The y- axis is set in decimal increments ,equal to function values of the corresponding angles.

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Maximum and Minimum value of a function

Maximum and minimum value of a function can be determined by simplifying the function with given intervals of time.

Function is a concept which expresses the idea that one quantity completely determines another quantity. Here is one such problem for your practice and understand how the value of function varies with respect to given interval of time.

Question : Find the maximum and minimum value of the function y = x³ - x on the interval [-3,3]

On substituting the value of x as -3 and 3, maxima and minima of function are found.

Solution :

When x = -3

y = x³ - x

= (-3)³ - (-3)

= -27 + 3

= - 24

And when x = 3

y = (3)³ - 3

= 27 - 3

= 24

Hence maximum value of function is 24 and minimum is - 24

A simple problem on ratio's

Topic:- Ratio's

If we wish to divide a amount or a number based on specific parts we use ratios.This will help us to share the values to different people in different quantities.

Here is an example which explain this better.

Question:-

How to share 35 between two people with the ration of 1:6

Answer:-

The given ratio is 1:6

The sum of the ratios is 1+6=7

Let us assume it is divided between A and B

    1
A= ----- * 35
    7

  35
= -----
  7

=  5

    6
B= ----- * 35
    7

   210
= -------
    7

So A gets 5 and b gets 30

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Problem on Mixed fractions

Topic:-Mixed fractions

A fraction is a number that can represent part of a whole.
The earliest fractions were reciprocals of integers, symbols representing one half, one third, one quarter, and so on.A much later development were the common or vulgar fractions which are still used today, and which consist of a numerator and a denominator.

In this algebra help ,we are giving an example of mixed fraction problem.
Topic:- Mixed Fractions
Question:-

 
 
  3      10
8--- - 4---- = ? 
  5      15


Answer:-

  3      10
8--- - 4---- = ? 
  5      15

(5*8)+3      (15*4)+10
--------  -  ----------
  5              15

 43          70
------  -  ------- 
  5          15

Multiply the numarator and the 
denominator of the first fraction 
with 3 to make equal denomitors 


   43*3         70
 -------  -   ------
   5*3          15

 
    129       70
   -----  -  -----
    15        15

      129-70
    -----------
        15

          59        14
         ----  or 3------
          15        15

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Examples of Locus

Topic:- Locus

Locus refers to a collection of points.

It's generally used to define a figure

Example 1:


A circle is a locus of all points ,Which are equidistant from a fixed point ,called the center of the circle.


Example 2:

A line is the locus of all points equidistant from two fixed points or from two parallel lines.

Let us consider a line

C_____________________________D >

-   -   -   -   -   -  -
A_____________________________B >
 k  l  m  o  n  p  q

Here,AB is a collection of points k,l,m,n,o,p,q
So,AB is called as Locus of all these points.

Commom point:
All the points are equidistant from line CD.

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