Answer to Question #118434 in Complex Analysis for Nii Laryea

Question #118434
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.
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Expert's answer
2020-06-01T18:05:32-0400
w=az+bz+cw={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+b3i+cw=3i={-3ai+b\over -3i+c}9+3ci=3ai+b=>b=9,a=c9+3ci=-3ai+b=>b=9, a=-c

w=14i=a(1+4i)+91+4iaw=1-4i={a(1+4i)+9\over 1+4i-a}17a+4ai=a+4ai+917-a+4ai=a+4ai+9a=4,b=9,c=4a=4, b=9, c=-4

w=4z+9z4w={4z+9\over z-4}

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


zˉ=4z+9z4\bar{z}={4z+9\over z-4}

x2+y24(xiy)4(x+iy)9=0x^2+y^2-4(x-iy)-4(x+iy)-9=0

x28x+y29=0x^2-8x+y^2-9=0

x28x+1616+y29=0x^2-8x+16-16+y^2-9=0

(x4)2+y2=25(x-4)^2+y^2=25

This is the cartesian equation of the circle with center (4,0)(4,0) and radius 5.5.


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

z=4+iyz=4+iy


4(4+iy)+94+iy4=25+4iyiy=4i(25y),y0{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.z=4.



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