The number of points z in the argand plane which satisfy the conditions Re z−2z−4i=0 and Imz−2z−4i=1 simultaneously is

GivenRe⁡(z−2z−4i)=0and in(z−2z−4i)=1 z−2z−4i=1 z−2=iz+4 z(1−i)=6⇒z=61−i=61−i×1+i1+i=3(1+i)∴only one z in argand plane.

The number of points z in the argand plane which satisfy the conditions Re z−2z−4i=0 and Imz−2z−4i=1 simultaneously is

GivenRe⁡(z−2z−4i)=0and in(z−2z−4i)=1 z−2z−4i=1 z−2=iz+4 z(1−i)=6⇒z=61−i=61−i×1+i1+i=3(1+i)∴only one z in argand plane.

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The number of points z in the argand plane which satisfy the conditions Re $\begin{array}{l} (\frac{z - 2}{z - 4 i}) = 0 \end{array}$ and $Im (\frac{z - 2}{z - 4 i}) = 1$ simultaneously is

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Infinitely many

answer is B.

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Detailed Solution

$Given Re (\frac{z - 2}{z - 4 i}) = 0 and in (\frac{z - 2}{z - 4 i}) = 1$
$\begin{aligned} \frac{z - 2}{z - 4 i} = 1 \\ z - 2 = i z + 4 \\ z (1 - i) = 6 \end{aligned}$
$\Rightarrow z = \frac{6}{1 - i} = \frac{6}{1 - i} \times \frac{1 + i}{1 + i} = 3 (1 + i)$
$∴$ only one z in argand plane.