Question

If $ |{{z}^{2}}-1|=|z{{|}^{2}}+1, $ then z lies on

• A. the real axis
• B. the imaginary axis
• C. a circle
• D. an ellipse

Answer 1

Answer: (B)

Given that $ |{{z}^{2}}-1|=|z{{|}^{2}}+1 $ $ \Rightarrow $ $ |{{z}^{2}}+(-1)|=|{{z}^{2}}|+|-1| $ It shows that the origin, $ -1 $ and $ {{z}^{2}} $ lies on a line and $ {{z}^{2}} $ and $ -1 $ lies on one side of the origin, therefore $ {{z}^{2}} $ is a negative number. Hence z will be purely imaginary. So we can say that z lies on y-axis. Alternate Solution We know that, if $ |{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}| $ $ \Rightarrow $ $ \arg ({{z}_{1}})=\arg ({{z}_{2}}) $ $ \because $ $ |{{z}_{2}}+(-1)|=|{{z}^{2}}|+|-1| $ $ \Rightarrow $ $ \arg ({{z}^{2}})=\arg (-1) $ $ \Rightarrow $ $ 2\arg (z)=\pi $ $ (\because \arg (-1)=\pi ) $ $ \Rightarrow $ $ \arg (z)=\frac{\pi }{2} $ $ \Rightarrow $ $ z $ lies on y-axis (imaginary axis).