1.

if z lies on |z| = 2, then \( \rm \left|\dfrac 4 {\bar z} \right|\) lies onWhere \(\rm \bar z\) is the conjugate of a complex number1. a circle2. an ellipse3. a straight line 4. None of these

Answer» Correct Answer - Option 1 : a circle

Concept:

 Let z  = x + iy, then \( {\bar z} \) is the conjugate of a complex number given by,

\(\rm {\bar z} = x - iy\)

Calculations:

Let z  = x + iy lies on |z| = 2

⇒ \(\rm \sqrt {x^2+y^2} = 2\)

⇒ \(\rm {x^2+y^2} = 4\), which is a circle with centre at (0, 0) and radius is 2.

 Let z  = x + iy, then \( {\bar z} \) is the conjugate of a complex number given by,

\(\rm {\bar z} = x - iy\)

Now, \(\rm \dfrac 4 {\bar z} = \dfrac {4}{x -iy}\)

 \(⇒ \rm \dfrac 4 {\bar z} = \dfrac {4}{x -iy} \times \dfrac {x+iy}{x+iy}\)

\(⇒ \rm \dfrac 4 {\bar z} = \dfrac {4(x+iy)}{x^2+y^2}\)
\(⇒ \rm \dfrac 4 {\bar z} = \dfrac {4z}{4}\)          (∵ \(\rm {x^2+y^2} = 4\))
\(⇒ \rm \dfrac 4 {\bar z} = z\)

Taking mod both sides, we get

\(⇒ \rm \left|\dfrac 4 {\bar z} \right|= |z|\)

Let \( \rm \dfrac 4 {\bar z} = z'\)

Hence, |z'| = 2            (∵  |z| = 2)

|z'| lies on the circle.


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