w=z+caz+b 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=−3i+c−3ai+b9+3ci=−3ai+b=>b=9,a=−c
w=1−4i=1+4i−aa(1+4i)+917−a+4ai=a+4ai+9a=4,b=9,c=−4
w=z−44z+9(a) Show that the points for which z is transformed to zˉ lie on a circle and give the centre and radius of this circle.
zˉ=z−44z+9
x2+y2−4(x−iy)−4(x+iy)−9=0
x2−8x+y2−9=0
x2−8x+16−16+y2−9=0
(x−4)2+y2=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+iy−44(4+iy)+9=iy25+4iy=4−i(y25),y=0 Every point of the line is transformed to point of this line excepting point z=4.
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