I spent hours on this question. couldn't do. Points A, B and C represent i, 3-i and 4+2i in....

an Argand Diagram. D is the reflection of C in the line AB. Find the complex number which is represented by D.

i can sure make the diagram. but please tell me what to do with it. i can't solve the question. thanks.

You want to reflect the point (4,2) about the line between (0,1) and (3,-1).

The point of reflection will be where these two lines meet:

y = 3/2 x - 4
y = -2/3 x + 1

(Since the slope of the line from A to B is -2/3, the slope of the perpendicular line will be 3/2 - we want the line going through (4,2) with slope 3/2.)

-2/3 x + 1 = 3/2 x -4
5 = 13/6 x
x = 30/13

And the reflected point will be equidistant on the opposite side:

x = 8/13
y = -3 1/13

The answer is (8/13, -40/13) = 8/13 - 40/13 i

Line from A(0,1) to B(3,−1) is
y = −2/3 x + 1
2x + 3y − 3 = 0

Line from C(4,2) to D(a,b) has slope = 3/2
(b−2)/(a−4) = 3/2
b = 3/2 a − 4

Perpendicular distance from D to line AB = perpendicular distance from C to line AB
|2a + 3b − 3| / √(4+9) = |2(4) + 3(2) − 3| / √(4+9)
|2a + 3b − 3| = 11
2a + 3(3/2 a − 4) − 3 = ± 11
13/2 a − 15 = ± 11
13/2 a = 15 ± 11
a = 2 (15 ± 11) / 13

a = 2 (15 + 11) / 13 = 4 -----> x-coordinate (real coordinate) of C
a = 2 (15 − 11) / 13 = 8/13

b = 3/2 a − 4 = 3/2 (8/13) − 4 = 12/13 − 52/13 = −40/13

Point D on Argand Diagram = (8/13 − 40/13)

Complex number represented by D: 8/13 − 40/13 i