The equation of the line joining the complex numbers -5 + 4i and 7 + 2i can be expressed in the form az + b \overline{z} = 38 for some complex numbers a and b. Find the product ab.
I can get the equation of the line by slope intercept but I don't know how to do this in the form az + b \overline{z} = 38
To express the equation of the line in the form az + b \overline{z} = 38, we need to find the values of a and b.
Let's start by finding the slope of the line. The slope is calculated as the difference in the imaginary parts divided by the difference in the real parts between the two given complex numbers.
Given:
Complex number 1: -5 + 4i
Complex number 2: 7 + 2i
Difference in imaginary parts = 2i - 4i = -2i
Difference in real parts = 7 - (-5) = 12
Therefore, the slope is -2i / 12 = -1/6i.
Now, we need to find the values of a and b. The equation az + b \overline{z} = 38 represents a line in the complex plane. The term az represents the line's slope, and the term b \overline{z} represents the complex conjugate of z.
Rewriting the equation in terms of the given complex numbers, we get:
a(-5 + 4i) + b(-5 - 4i) = 38
Expanding this equation gives:
(-5a - 5b) + (4ai - 4bi) = 38
Separating the real and imaginary parts of the equation, we have:
-5(a + b) + 4(a - b)i = 38
From this equation, we can equate the real and imaginary parts separately:
-5(a + b) = 38 (Equation 1)
4(a - b) = 0 (Equation 2)
From Equation 2, we can see that a - b must be equal to zero. This implies that a = b.
Substituting a = b into Equation 1, we get:
-5(2a) = 38
-10a = 38
a = -38 / 10
a = -19 / 5
Since a = b, we have b = -19 / 5 as well.
Finally, the product ab is given by:
ab = (-19 / 5) * (-19 / 5)
ab = 361 / 25
Hence, the product ab is equal to 361 / 25.