Complex Numbers And Quadratic Equations question 354
Question: When $ \frac{z+i}{z+2} $ is purely imaginary, the locus described by the point $ z $ in the Argand diagram is a
Options:
A) Circle of radius $ \frac{\sqrt{5}}{2} $
B) Circle of radius $ \frac{5}{4} $
C) Straight line
D) Parabola
Show Answer
Answer:
Correct Answer: A
Solution:
Given that Im $ ( \frac{z+i}{z+2} ) $ Let $ z=x+iy $ Þ $ \frac{x+iy+i}{x+iy+2} $ = $ \frac{x+i(y+1)}{(x+2)+iy} $ $ =\frac{[x+i(y+1)][(x+2)-iy]}{[(x+2)+iy][(x+2)-iy]} $ $ =[ \frac{x^{2}+2x+y^{2}+y}{{{(x+2)}^{2}}+y^{2}} ]+i[ \frac{(y+1)(x+2)-xy}{{{(x+2)}^{2}}+y^{2}} ] $ If it is purely imaginary then real part must be equal to zero. Þ $ \frac{x^{2}+y^{2}+2x+y}{{{(x+2)}^{2}}+y^{2}}=0 $ Þ $ x^{2}+y^{2}+2x+y=0 $ Which is a circle and its radius is given by $ \sqrt{g^{2}+f^{2}-c}=\sqrt{1+\frac{1}{4}-0}=\frac{\sqrt{5}}{2} $ Therefore Argand diagram is circle of radius $ \frac{\sqrt{5}}{2} $ .
BETA
