Find the value of x and y in the following (i) (2+3 i)(4-7 i)=x+i y (ii) ((2 x+4 i)(1+i))/((3-2

Find the value of x and y in the following (i) (2+3 i)(4-7...

Key Concepts

Complex Numbers

Complex numbers are expressions in the form a + i·b, where a and b are real numbers and i is the imaginary unit with the property that i² = -1. They extend the concept of one-dimensional real numbers to the two-dimensional complex plane, allowing solutions to equations that have no real solutions.

Arithmetic Operations with Complex Numbers

This concept covers the basic operations on complex numbers, including addition, subtraction, multiplication, and division. Multiplication of complex numbers involves distributing each term and using the identity i² = -1, whereas division typically requires multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Rationalizing Complex Denominators

When dividing by a complex number, it is often necessary to convert the denominator into a real number by multiplying both the numerator and the denominator by the complex conjugate of the denominator. This technique simplifies the expression and isolates the real and imaginary components.

Equating Real and Imaginary Parts

This method involves setting the real parts equal to each other and the imaginary parts equal to each other in an equation involving complex numbers. It is a fundamental technique used to solve for unknowns by converting a complex equation into two separate real equations.

Exponentiation of Complex Numbers

Computing powers of complex numbers can be achieved through algebraic expansion or by using polar form and De Moivre's Theorem. This approach simplifies the process of raising a complex number to a power by converting multiplication into a form that involves the magnitude and angle of the complex number.

Transcript

00:01 So, in the given question, we are told to find the x and y values of the following complex numbers that are given in the question.

00:10 So the first complex number that is given in the question is 2 plus 3i times 4 minus 7i is equal to x plus i y.

00:27 So let's start by multiplying this complex numbers.

00:30 So we will have 8 plus 12 i minus 14 i plus 21 i square i square is equal to minus 1 since i is the complex square root the complex square root of minus 1 right so it is equal to we can write it as the square root of minus 1.

01:03 It is a complex number which is the square root of minus 1.

01:08 So i square is equal to 1 minus 1.

01:11 And now we can write 8 plus 12 i minus 14 i is minus 2 i plus 21 times i square is 21 times minus 1 which is minus 21 and this is equal to 29 minus 2 and from this we can write the value of x is equal to 20 in i and the value of y is equal to minus 2.

01:45 Now let's look at the second complex number that is given in the question and it is given as 2x plus 4 i times 1 plus i divided by 3 minus 2 i so this is equal to 2 2 2 i.

02:10 Plus iy.

02:12 So this is what is given in the question.

02:15 So what we do is we can take let's multiply in both numerator and denominator the complex number 3 plus 2y and on doing that on the left hand side of this complex number what we will have is we can simplify the denominator.

02:35 Right so what we will have is numerator times 3 plus 2y divided by denominator times 3 plus 2y and this is equal to this is equal to we will get the simplified answer by multiplying these two in the denominator and these two in the denominator we will have two.

03:07 2x plus 4 i times 3 plus 3 i plus 2 i plus 2 i squared divided by in the denominator it is of the form a plus b times a minus b which is equal to a square minus b square so we can take 3 square minus 2 i square and this is equal to this is equal to 2x plus 4 i times times 1 plus 5 i because 2 i becomes minus 2 since i square is equal to minus 1 so 1 plus 5 i divided by 3 square minus 2 is is equal to 13.

04:03 And now what we can do is to write the real part and the imaginary part separately in the numerator after multiplying these numbers.

04:17 So we will have 2x plus 4i plus 10 ix plus 20 i squared divided by 13 which would be equal to 2x minus 20 minus because we have i square plus i times 4 plus 10x and this is divided by 13 right so now we can equate the term in the left hand side of this equation with the complex number that was given in the right hand side of the equation as 2 plus iy and what we can write is 2x minus 20 divided by 13 which is the real part is equal to the real part on the right hand side which is 2 and this would give us 2x minus 20 is equal to 26 and 2x is equal to 4x is equal to 4 ,000.

05:27 Which means x is equal to 46 by 2 which is 23 and if x is equal to 23 we can now take the imaginary part on the left hand side which is 4 plus 10 x divided by 13 which is equal to y in the right hand side so we will have 4 plus 10 times 23 divided by 13 is equal to y 10 times 23 is 230 so 234 divided by 13 is equal to is equal to 18 which is equal to y so the x value is 23 and y value is 18 in the second complex number that is given in the question so now let's move on to the third complex number which is given as 1 minus 3 i whole square is equal to x plus iy so we will have 1 square minus 6 i squared plus 3 i squared is equal to x plus iy and from this we can write 1 minus over here it is not i square it is 6 i so 1 minus 6 i minus 6 i minus 3 square is 9 so minus 9 is equal to 1 minus 9 is minus 8 so minus 9 is minus 8 so minus 9...

Please give Ace some feedback