Answer:
The answer is "[tex]\boxed{z= \frac{-16}{5}- \frac{-12}{15}i}[/tex]".
Explanation:
As the problem stands
At the point of P, it is the complex number z in the Diagram of Argand and z = X+iy.
We have said this: [tex](z+2)= \lambda i (z +8) .... (i)[/tex]
where the [tex]\lambda[/tex] parameter is a true
The conceptual equation of the locus P varies between [tex]z = x+iy \ \ \ to \ \ \ \lambda[/tex]
And in equation mentioned above.
[tex]x+iy+2=\lambda i(x+iy+8) \\\\x+iy+2= \lambda xi+ \lambda i^2y+\lambda 8i\\\\ x+2+iy=-y \lambda +i(x+8)\lambda\\\ compare \ real \ and \ imaginary\ part \\\\\ x+2 = -y\lambda \\\\y= (x+8) \lambda\\\\ \lambda = \frac{x+2}{-y} \\ \\ \lambda = \frac{y}{x+8}[/tex]
[tex]y^2= -x^2-10x-16 ....(ii)\\\\z= \mu (4+3i)....(iii)\\\\\ z= x+iy \\\\x+iy = 4\mu + 3 \mu i \\\\x= 4\mu \\\\y= 3\mu[/tex]
put the value of x, y in equation (ii) we get:
[tex]5\mu +4=0\\\\\mu = \frac{-4}{5} \\\\[/tex]
to put the of [tex]\mu[/tex] in equation (iii) we get:
[tex]z= \frac{-4}{5} (4+3i) \\\\ \boxed{z= \frac{-16}{5}- \frac{-12}{15}i} \\[/tex]
