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AP EAMCET Engineering 20 Apr 2019 Shift 2 Paper

Section: Mathematics

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Question : 8 of 160

Marks: +1, -0

In the complex plane C. the set

\{ z \in C : \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4} \}

represents

a straight line
a circle
a parabola
an ellipse

Solution:

It is given that,

\{ z \in C : \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4} \}

Let the complex number be

z = x + i y

Thus,

\frac{z-1}{z+1} = \frac{(x-1)+i y}{(x+1)+i y} \times \frac{(x+i y)-i y}{(x+i y)-i y}

= \frac{x^2 + y - 1^2}{(x+1)^2 + y^2} + i\left(\frac{2y}{(x+1)^2 + y^2}\right)

Therefore

\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}

\left(\frac{2y}{x^2 + y^2 - 1}\right) = 1

x^2 + y^2 - 2y - 1 = 0

The complex plane represents a circle.

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