Let L be the line whose equation is
Re(z) + Im(z) = 2.
Find the equation of the inversion of L, expressed in terms of a complex variable z.
Thanks so much!
Lexi
To find the equation of the inversion of the line L, we first need to understand what inversion means in complex analysis. Inversion is a transformation that maps a point z in the complex plane to a new point w such that the product of their distances from a fixed center point is equal to the square of the radius of that fixed center point.
Let's denote the center of inversion as c and the radius as r. The equation of the inversion of a point z is given by:
w = c + (r^2 / (z - c))
To find the equation of the inversion of the line L, we need to express the equation of the line L in complex form.
The equation of line L is given by:
Re(z) + Im(z) = 2
To express this equation in complex form, we can rewrite it as:
z + conj(z) = 2
where conj(z) represents the complex conjugate of z.
Now, we can apply the equation of inversion to find the equation of the inversion line, denoted as M:
w = c + (r^2 / (z - c))
Plugging in the equation of line L, we have:
w = c + (r^2 / (z - c)) = z + conj(z)
Simplifying this equation, we get:
c + (r^2 / (z - c)) = z + conj(z)
To isolate w, multiply through by (z - c):
c(z - c) + r^2 = (z + conj(z))(z - c)
Expanding both sides gives us:
cz - c^2 + r^2 = z^2 - cz + z*conj(z) - c*conj(z)
Rearranging the terms and combining like terms, we get:
z^2 - (c + conj(c))z + (c^2 + r^2 - c*conj(c)) = 0
Now, we have the equation of the inversion line M expressed in terms of z:
z^2 - (c + conj(c))z + (c^2 + r^2 - c*conj(c)) = 0
This is the equation of the inversion of the given line L, expressed in terms of the complex variable z.
Please note that the specific values of c and r are not given in the original question, so you'll need to find them separately if necessary.