Let z - x - iy where x- y are real variables and i- -1- If -2z-1- - -z-2- then the point z describes -

The correct option is C A circleGiven: z=x+iy ⇒ 2z 1 = 2x 1 + 2iy #160;and #160; z 2=x 2 + iy Given, | 2z 1| = | z 2 | ⇒√((2x 1) ...

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Byju's Answer

Standard XII

Mathematics

Modulus of a Complex Number

Let z = x +...

Question

Let $z = x + i y$ where x, y are real variables and $i = \sqrt{- 1}$ . If $| 2 z - 1 | = | z - 2 |$ , then the point z describes :

A circle

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An ellipse

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A hyperbola

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A parabola

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Solution

The correct option is C A circle
Given: $z = x + i y$
$\Rightarrow 2 z - 1 = 2 x - 1 + 2 i y$ and $z - 2 = x - 2 + i y$

Given, $| 2 z - 1 | = | z - 2 |$
$\Rightarrow \sqrt{(2 x - 1)^{2} + (2 y)^{2}} = \sqrt{(x - 2)^{2} + y^{2}}$
$\Rightarrow (2 x - 1)^{2} + (2 y)^{2} = (x - 2)^{2} + y^{2}$
$\Rightarrow x^{2} + y^{2} = 1$

This is a locus for circle, hence the point $z$ describes a circle.

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Let z=x+iy where x, y are real variables and i=√−1. If |2z−1|=|z−2|, then the point z describes :

The correct option is C A circleGiven: z=x+iy⇒2z−1=2x−1+2iy and z−2=x−2+iyGiven, |2z−1|=|z−2|⇒√(2x−1)2+(2y)2=√(x−2)2+y2⇒(2x−1)2+(2y)2=(x−2)2+y2⇒x2+y2=1This is a locus for circle, hence the point z describes a circle.

Let $z = x + i y$ where x, y are real variables and $i = \sqrt{- 1}$ . If $| 2 z - 1 | = | z - 2 |$ , then the point z describes :