MCQ
A complex number $z$ is such that $arg\,\left( {\frac{{z - 2}}{{z + 2}}} \right)$ $ = \frac{\pi }{3}$. The points representing this complex number will lie on
- AAn ellipse
- BA parabola
- ✓A circle
- DA straight line
✓
Answer
Correct option: C.
A circle
c
(c) $arg\,\,\,\left( {\frac{{z - 2}}{{z + 2}}} \right) = \frac{\pi }{3}$ $ \Rightarrow $${\tan ^{ - 1}}\left[ {\frac{{(x - 2) + iy}}{{(x + 2)\, + iy}}} \right] = \frac{\pi }{3}$
(c) $arg\,\,\,\left( {\frac{{z - 2}}{{z + 2}}} \right) = \frac{\pi }{3}$ $ \Rightarrow $${\tan ^{ - 1}}\left[ {\frac{{(x - 2) + iy}}{{(x + 2)\, + iy}}} \right] = \frac{\pi }{3}$
$ \Rightarrow $ $\sqrt {{{(x - 2)}^2} + {y^2}} = \tan (\pi /3)\,[\sqrt {{{(x + 2)}^2} + {y^2}} ]$
Squaring both sides,
$ \Rightarrow $ ${(x - 2)^2} + {y^2} = 3{[x + 2]^2} + {y^2}]$
$ \Rightarrow $ ${x^2} + {y^2} + 4 - 4x = 3{x^2} + 3{y^2} + 12x + 12$
$ \Rightarrow $ $2{x^2} + 2{y^2} + 16x + 8 = 0$
$ \Rightarrow $${x^2} + {y^2} + 8x + 4 = 0$
which is a equation of circle.
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