Example for complex conjugates

In grade 10 math , complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of opposite signs. For example, 3 + 4i and 3 - 4i are complex conjugates.
The conjugate of the complex number z

where a and b are real numbers, is

An alternate notation for the complex conjugate is z * . However, the notation avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of complex conjugation. If a complex number is represented as a 2×2 matrix, the notations are identical.We also can use scientific notation converter to find the number.

Let's an example problem from numeric and algebraic operations


Question:-

How do you use complex conjugates to find (3+7i)/(2-i) ?


Answer:-


In the Given problem the denominator is (2-i)

So ,it's conjugate is (2+i)

Multiplying both numerator and denominator with (2+i)

= (3+7i)(2+i)/(2-i)(2+i)

= (5+17i)/(4-i²)

We know that i² = -1

= (5+17i)/4+1

= (5+17i)/5 Answer

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

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

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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