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Question

Mathematics Question on Complex Numbers and Quadratic Equations

If z is a complex number then ∣z+1∣=3∣z−1∣|z+1|=\sqrt3|z-1| represents

A

Straight line

B

Ellipse

C

Hyperbola

D

circle

Answer

circle

Explanation

Solution

∣z+1∣2=3∣z−1∣2\left|z+1\right|^{2}=3\left|z-1\right|^{2} ⇒(x+1)2+y2=3[(x−1)2+y2]\Rightarrow \left(x+1\right)^{2}+y^{2}=3\left[\left(x-1\right)^{2}+y^{2}\right] ⇒2(x2+y2)−8x+2=0\Rightarrow 2\left(x^{2}+y^{2}\right)-8x+2=0 ⇒x2+y2−4x+1=0\Rightarrow x^{2}+y^{2}-4x+1=0 which is a circle.