Find the complex number which satisfied the equation 5zz ̅+3(z-z ̅)=42 – 15j
5(x+iy)(x-iy)+3(x+iy -x +iy) =42-15i
5x^2+5y^2 + 6iy = 42-15i
5 x^2 + 5 y^2 = 42
6 i y = -15i
y = -15/6
5 x^2 + 5 (225/36) = 42
To find the complex number that satisfies the given equation, let's break it down step by step.
First, let's expand the expression. We know that the complex conjugate of a complex number z is denoted as z ̅ and is formed by changing the sign of the imaginary part. Therefore, we can expand the equation as follows:
5zz ̅ + 3(z - z ̅) = 42 - 15j
Expanding further:
5zz ̅ + 3z - 3z ̅ = 42 - 15j
Next, let's separate the real and imaginary parts of the equation. The real part is represented by Re( ) and the imaginary part by Im( ). We can create two equations by equating the real and imaginary parts separately:
Real Part: 5zz ̅ + 3z - 3z ̅ = 42
Imaginary Part: 0 = -15j
Let's solve the imaginary part equation first. We know that the imaginary part of a complex number is zero when the complex number is purely real. Therefore, we can conclude that the complex number z is purely real.
Substituting this information into the real part equation:
5zz ̅ + 3z - 3z ̅ = 42
Since z is real, we have z - z ̅ = 0. Therefore, the equation simplifies to:
5z * z ̅ = 42
Now, we have a simplified equation in terms of z and z ̅. We can solve this equation by substituting z = a + bi and z ̅ = a - bi, where a and b are real numbers.
(5(a + bi)(a - bi)) = 42
Expanding:
5(a^2 - b^2i^2) = 42
Since i^2 = -1, the equation becomes:
5(a^2 + b^2) = 42
Now, we can solve for a and b by dividing both sides of the equation by 5:
a^2 + b^2 = 42/5
This equation represents the equation of the unit circle in the complex plane, where the real part a corresponds to the x-coordinate and the imaginary part b corresponds to the y-coordinate.
To find the possible values of a and b, we need to determine the radius of the unit circle. The radius is given by the square root of the right-hand side of the equation:
r = √(42/5)
Now, we can express a and b in polar form, where r represents the modulus of the complex number and θ represents the argument:
a = r * cos(θ)
b = r * sin(θ)
Plugging in the values:
a = √(42/5) * cos(θ)
b = √(42/5) * sin(θ)
Therefore, the complex number z satisfies the equation 5zz ̅+3(z-z ̅)=42 – 15j when z = a + bi, where a = √(42/5) * cos(θ) and b = √(42/5) * sin(θ). The value of θ can vary between 0 and 2π, representing all the possible solutions on the unit circle.