Question

The transformation T : z "→" w in the complex plane is defined by w = (az + b)/( z + c), where a, b, c ∈ R. Given that w = 3i when z = −3i, and w = 10 − 4i when z = 1 + 4i, find the values of a, b and c. (a) Show that the points for which z is transformed to z lie on a circle and give the centre and radius of this circle. (b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

Answer 1

$$w={az+b\over z+c}$$

Given that w = 3i when z = −3i, and w = 1 − 4i when z = 1 + 4i, find the values of a, b and c. 

$$w=3i={-3ai+b\over -3i+c}$$ $$9+3ci=-3ai+b=>b=9, a=-c$$

$$w=1-4i={a(1+4i)+9\over 1+4i-a}$$ $$17-a+4ai=a+4ai+9$$ $$a=4, b=9, c=-4$$

$$w={4z+9\over z-4}$$

(a) Show that the points for which $z$ is transformed to $\bar{z}$ lie on a circle and give the centre and radius of this circle. 

$$\bar{z}={4z+9\over z-4}$$

$$x^2+y^2-4(x-iy)-4(x+iy)-9=0$$

$$x^2-8x+y^2-9=0$$

$$x^2-8x+16-16+y^2-9=0$$

$$(x-4)^2+y^2=25$$

This is the cartesian equation of the circle with center $(4,0)$ and radius $5.$

(b) Show that the line through the point z = 4 and perpendicular to the real axis is invariant under T.

$z=4+iy$

$${4(4+iy)+9\over 4+iy-4}={25+4iy\over iy}=4-i({25\over y}), y\not=0$$

Every point of the line is transformed to point of this line excepting point $z=4.$