Question

ABCD is a parallelogram oriented anticlockwise and CDEF is a square oriented clockwise. If A = 2 − 3i, B = 3 + 4i, C = −3 − i, find AE² .

Answer 1

First we have to find coordinates of D.

It will be D = (-4, -8)

(∵ AB = CD & AD = BC for D = (-4, -8))

Now, we have to find coordinates of E.

\(\because DE \perp CD.\) 

Let D = (x, y).

\(\left(\frac {y +8}{x+4}\right) \left(\frac{-1+8}{-3+4}\right) = -1\)

⇒ \(7(y+ 8) = - (x +4)\)  ......(1)

Also,

\(DE = CD\)

⇒ \((y + 8)^2 + (x + 4)^2 = (-3 + 4)^2 + (-1 + 8)^2\)

⇒ \((y +8)^2 + 49(y +8)^2 = 1 +49\)

⇒ \((y + 8)^2 = 1\)

⇒ \(y +8 = 1\)

⇒ \(y = -8 + 1 = -7\)

\(\therefore x = -7(y +8)-4\)

\(= -7 (-7 + 8)-4\)

\(= -7 -4 \)

\(= -11\)

\(\therefore E =(-11, -7)\)

\(\therefore AE^2 = (2 +11)^2 + (-3 +7)^2\)

\(= 13 ^2 + 4^2 = 169 + 16 = 185\)