A transformation T T T z z z z = x + i y z = x + iy z = x + i y w w w w = u + i v w = u + iv w = u + i v
w = z − 3 2 i − z z ≠ 2 i \begin{align*}
w = \frac{z - 3}{2\mathrm{i} - z} \qquad z \neq 2\mathrm{i}
\end{align*} w = 2 i − z z − 3 z = 2 i The line in the z z z y = x + 3 y = x + 3 y = x + 3 T T T C C C w w w
(a) Determine
(i) the coordinates of the centre of C C C
(ii) the exact radius of C C C
(8)
The region y > x + 3 y > x + 3 y > x + 3 z z z T T T R R R w w w
(b) On a single Argand diagram
(i) sketch the circle C C C
(ii) shade and label the region R R R
(2)
解答
(a)
The transformation is given by:
w = z − 3 2 i − z \begin{align*}
w = \frac{z-3}{2\mathrm{i}-z}
\end{align*} w = 2 i − z z − 3 Rearrange to make z z z
w ( 2 i − z ) = z − 3 2 i w − w z = z − 3 z + w z = 2 i w + 3 z ( w + 1 ) = 3 + 2 i w z = 3 + 2 i w w + 1 \begin{align*}
w(2\mathrm{i}-z)
= &\,z - 3\\[4mm]
2\mathrm{i}w - wz
= &\,z - 3\\[4mm]
z + wz
= &\,2\mathrm{i}w + 3\\[4mm]
z(w+1)
= &\,3 + 2\mathrm{i}w\\[4mm]
z
= &\,\frac{3+2\mathrm{i}w}{w+1}
\end{align*} w ( 2 i − z ) = 2 i w − w z = z + w z = z ( w + 1 ) = z = z − 3 z − 3 2 i w + 3 3 + 2 i w w + 1 3 + 2 i w Let z = x + i y z = x + \mathrm{i}y z = x + i y w = u + i v w = u + \mathrm{i}v w = u + i v
x + i y = 3 + 2 i ( u + i v ) u + i v + 1 = ( 3 − 2 v ) + i ( 2 u ) ( u + 1 ) + i v \begin{align*}
x + \mathrm{i}y
= &\,\frac{3 + 2\mathrm{i}(u+\mathrm{i}v)}{u+\mathrm{i}v+1}\\[4mm]
= &\,\frac{(3-2v) + \mathrm{i}(2u)}{(u+1) + \mathrm{i}v}
\end{align*} x + i y = = u + i v + 1 3 + 2 i ( u + i v ) ( u + 1 ) + i v ( 3 − 2 v ) + i ( 2 u ) Multiply the numerator and denominator by the complex conjugate of the denominator, ( u + 1 ) − i v (u+1) - \mathrm{i}v ( u + 1 ) − i v
x + i y = [ ( 3 − 2 v ) + i ( 2 u ) ] [ ( u + 1 ) − i v ] ( u + 1 ) 2 + v 2 = ( 3 − 2 v ) ( u + 1 ) + 2 u v + i [ 2 u ( u + 1 ) − v ( 3 − 2 v ) ] ( u + 1 ) 2 + v 2 \begin{align*}
x + \mathrm{i}y
= &\,\frac{[(3-2v) + \mathrm{i}(2u)][(u+1) - \mathrm{i}v]}{(u+1)^2 + v^2}\\[4mm]
= &\,\frac{(3-2v)(u+1) + 2uv + \mathrm{i}[2u(u+1) - v(3-2v)]}{(u+1)^2 + v^2}
\end{align*} x + i y = = ( u + 1 ) 2 + v 2 [( 3 − 2 v ) + i ( 2 u )] [( u + 1 ) − i v ] ( u + 1 ) 2 + v 2 ( 3 − 2 v ) ( u + 1 ) + 2 uv + i [ 2 u ( u + 1 ) − v ( 3 − 2 v )] Equate the real parts (x x x y y y
x = ( 3 − 2 v ) ( u + 1 ) + 2 u v ( u + 1 ) 2 + v 2 = 3 u + 3 − 2 u v − 2 v + 2 u v ( u + 1 ) 2 + v 2 = 3 u − 2 v + 3 ( u + 1 ) 2 + v 2 y = 2 u 2 + 2 u − 3 v + 2 v 2 ( u + 1 ) 2 + v 2 \begin{align*}
x
= &\,\frac{(3-2v)(u+1) + 2uv}{(u+1)^2 + v^2} = \frac{3u + 3 - 2uv - 2v + 2uv}{(u+1)^2 + v^2} = \frac{3u - 2v + 3}{(u+1)^2 + v^2}\\[4mm]
y
= &\,\frac{2u^2 + 2u - 3v + 2v^2}{(u+1)^2 + v^2}
\end{align*} x = y = ( u + 1 ) 2 + v 2 ( 3 − 2 v ) ( u + 1 ) + 2 uv = ( u + 1 ) 2 + v 2 3 u + 3 − 2 uv − 2 v + 2 uv = ( u + 1 ) 2 + v 2 3 u − 2 v + 3 ( u + 1 ) 2 + v 2 2 u 2 + 2 u − 3 v + 2 v 2 We are given the line y = x + 3 y = x + 3 y = x + 3 y y y x x x
2 u 2 + 2 u − 3 v + 2 v 2 ( u + 1 ) 2 + v 2 = 3 u − 2 v + 3 ( u + 1 ) 2 + v 2 + 3 \begin{align*}
\frac{2u^2 + 2u - 3v + 2v^2}{(u+1)^2 + v^2}
= &\,\frac{3u - 2v + 3}{(u+1)^2 + v^2} + 3
\end{align*} ( u + 1 ) 2 + v 2 2 u 2 + 2 u − 3 v + 2 v 2 = ( u + 1 ) 2 + v 2 3 u − 2 v + 3 + 3 Multiply both sides by ( u + 1 ) 2 + v 2 (u+1)^2 + v^2 ( u + 1 ) 2 + v 2
2 u 2 + 2 u − 3 v + 2 v 2 = 3 u − 2 v + 3 + 3 [ ( u + 1 ) 2 + v 2 ] 2 u 2 + 2 u − 3 v + 2 v 2 = 3 u − 2 v + 3 + 3 u 2 + 6 u + 3 + 3 v 2 0 = u 2 + v 2 + 7 u + v + 6 \begin{align*}
2u^2 + 2u - 3v + 2v^2
= &\,3u - 2v + 3 + 3[(u+1)^2 + v^2]\\[4mm]
2u^2 + 2u - 3v + 2v^2
= &\,3u - 2v + 3 + 3u^2 + 6u + 3 + 3v^2\\[4mm]
0
= &\,u^2 + v^2 + 7u + v + 6
\end{align*} 2 u 2 + 2 u − 3 v + 2 v 2 = 2 u 2 + 2 u − 3 v + 2 v 2 = 0 = 3 u − 2 v + 3 + 3 [( u + 1 ) 2 + v 2 ] 3 u − 2 v + 3 + 3 u 2 + 6 u + 3 + 3 v 2 u 2 + v 2 + 7 u + v + 6 Complete the square for u u u v v v
( u + 7 2 ) 2 − 49 4 + ( v + 1 2 ) 2 − 1 4 + 6 = 0 ( u + 7 2 ) 2 + ( v + 1 2 ) 2 = 50 4 − 6 ( u + 7 2 ) 2 + ( v + 1 2 ) 2 = 13 2 \begin{align*}
\left(u + \frac{7}{2}\right)^2 - \frac{49}{4} + \left(v + \frac{1}{2}\right)^2 - \frac{1}{4} + 6
= &\,0\\[4mm]
\left(u + \frac{7}{2}\right)^2 + \left(v + \frac{1}{2}\right)^2
= &\,\frac{50}{4} - 6\\[4mm]
\left(u + \frac{7}{2}\right)^2 + \left(v + \frac{1}{2}\right)^2
= &\,\frac{13}{2}
\end{align*} ( u + 2 7 ) 2 − 4 49 + ( v + 2 1 ) 2 − 4 1 + 6 = ( u + 2 7 ) 2 + ( v + 2 1 ) 2 = ( u + 2 7 ) 2 + ( v + 2 1 ) 2 = 0 4 50 − 6 2 13 (i) The coordinates of the centre of C C C ( − 7 2 , − 1 2 ) \left(-\frac{7}{2}, -\frac{1}{2}\right) ( − 2 7 , − 2 1 ) (ii) The exact radius of C C C 13 2 = 26 2 \sqrt{\frac{13}{2}} = \frac{\sqrt{26}}{2} 2 13 = 2 26
(b)
(i) and (ii) :
The region y > x + 3 y > x + 3 y > x + 3 C C C w w w z = 4 i z = 4\mathrm{i} z = 4 i w w w C C C Sketch instruction : Draw an Argand diagram featuring a circle C C C ( − 7 2 , − 1 2 ) \left(-\frac{7}{2}, -\frac{1}{2}\right) ( − 2 7 , − 2 1 ) ≈ 2.55 \approx 2.55 ≈ 2.55 R R R
