The system of equations
|z - 2 - 2i| = \sqrt{23},
|z - 8 - 5i| = \sqrt{38}
has two solutions z1 and z2 in complex numbers. Find (z1 + z2)/2.
Im really just plain old confused. Could anyone help me out?
if z = x+yi, you have two equations:
(x-2)^2 + (y-2)^2 = 23
(x-8)^2 + (y-5)^2 = 38
They intersect at
(1/5(20±3√10), 3/5(5∓2√10))
check:
(1/5(20+3√10)-2)^2 + (3/5(5-2√10)-2)^2 = 23
You can verify the other.
To find the value of (z1 + z2)/2, we need to first find the solutions z1 and z2 for the given system of equations.
Let's break down the problem into steps:
Step 1: Determine the values of z for each equation
We have two equations given:
1) |z - 2 - 2i| = sqrt(23)
2) |z - 8 - 5i| = sqrt(38)
The notation |z - a - bi| represents the modulus of complex number z - a - bi, which is the distance between point z and point (a, b) in the complex plane.
Step 2: Solve each equation separately
For the first equation:
|z - 2 - 2i| = sqrt(23)
The modulus equation can be rewritten as a distance formula:
sqrt((Re(z) - 2)^2 + (Im(z) - 2)^2) = sqrt(23)
Square both sides of the equation to eliminate the square root:
(Re(z) - 2)^2 + (Im(z) - 2)^2 = 23
Expanding and rearranging the equation, we get:
(Re(z))^2 - 4Re(z) + 4 + (Im(z))^2 - 4Im(z) + 4 = 23
Combining like terms:
(Re(z))^2 + (Im(z))^2 - 4Re(z) - 4Im(z) - 15 = 0
This equation can be rewritten as:
|z|^2 - (4Re(z) + 4Im(z)) + 15 = 0
For the second equation:
|z - 8 - 5i| = sqrt(38)
Following the same steps as above, we can rewrite the equation as:
|z|^2 - (16Re(z) + 10Im(z)) + 89 = 0
Step 3: Solve the system of equations
Now we have two equations:
1) |z|^2 - (4Re(z) + 4Im(z)) + 15 = 0
2) |z|^2 - (16Re(z) + 10Im(z)) + 89 = 0
To solve this system of equations, we need to find the values of z that satisfy both equations simultaneously.
Step 4: Find the common solutions for both equations
One approach to solving the system is to equate the left sides and the right sides of both equations:
|z|^2 - (4Re(z) + 4Im(z)) + 15 = |z|^2 - (16Re(z) + 10Im(z)) + 89
Simplifying the equation, we get:
12Re(z) + 6Im(z) = 74
This equation represents a line in the complex plane, specifically the real part (Re(z)) and the imaginary part (Im(z)) of z that satisfy the equation.
Step 5: Find the values of z that satisfy the line equation
To find the values of z that satisfy this line equation, we need to substitute the equation for z into either of the original equations and solve for the other variable.
Let's substitute z = a + bi into the line equation, where a represents the real part of z and b represents the imaginary part of z:
12a + 6b = 74
Solving this linear equation, we find the values of a and b that satisfy it.
Once we have the values of a and b, we can find the corresponding values of z1 and z2, and then calculate (z1 + z2)/2.