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Home » Show that the complex number z, satisfying the condition arg ((z-1)/(z+1)) = π/4 lies on a circle.
Show that the complex number z, satisfying the condition arg ((z-1)/(z+1)) = π/4 lies on a circle.
CBSE | Class 11 | Complex Numbers and Quadratic Equations | Maths | NCERT Exemplar
Show that the complex number z, satisfying the condition arg ((z-1)/(z+1)) = π/4 lies on a circle.

Let

    \[z\text{ }=\text{ }x\text{ }+\text{ }iy\]

arg

    \[\left( \left( z-1 \right)/\left( z+1 \right) \right)\text{ }=\text{ }\pi /4\]

    \[\Rightarrow ~arg\text{ }\left( z\text{ }\text{ }1 \right)\text{ }\text{ }-arg\text{ }\left( z\text{ }+\text{ }1 \right)\text{ }=\text{ }\pi /4\]

    \[\Rightarrow ~arg\text{ }\left( x\text{ }+\text{ }iy\text{ }\text{ }1 \right)\text{ }\text{ }-arg\text{ }\left( x\text{ }+\text{ }iy\text{ }+\text{ }1 \right)\text{ }=\text{ }\pi /4\]

    \[\Rightarrow ~arg\text{ }\left( x+\text{ }\text{ }1\text{ }+\text{ }iy \right)\text{ }\text{ }-arg\text{ }\left( x\text{ }+\text{ }1\text{ }+\text{ }iy \right)\text{ }=\text{ }\pi /4\]

NCERT Exemplar Solutions For Class 11 Maths Chapter 5-14

    \[\Rightarrow ~{{x}^{2}}~+\text{ }{{y}^{2}}~\text{ }-1\text{ }=\text{ }2y\]

    \[\Rightarrow ~{{x}^{2}}~+\text{ }{{y}^{2}}~\text{ }-2y\text{ }\text{ }-1\text{ }=\text{ }0\]

The equation obtained represents the equation of a circle.

i

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