If the reflection of the line $\bar{a}z + a \bar{z}= 0 $ in the real axis is $\bar{\alpha}z + \alpha \bar{z}= 0 $ in the simplest form

Q: If the reflection of the line $\bar{a}z + a \bar{z}= 0 $ in the real axis is $\bar{\alpha}z + \alpha \bar{z}= 0 $ in the simplest form, then

(A) α + a is purely real

(B) $\bar{\alpha} – a $ is purely real

(C) $ \alpha -\bar{a}$ is purely real

(D) $ \bar{\alpha} -\bar{a}$ is purely imaginary

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Ans: (B) & (C)
Sol: Let a = a1 + ia2 and z = x + iy, then

$\bar{a}z + a \bar{z}= 0 $

⇒ a1 x + a2 y = 0

or , $\large y = (-\frac{a_1}{a_2})x $

Its reflection in the real axis (x-axis) is $\large y = (\frac{a_1}{a_2})x $

or, a1 x – a2 y = 0

i.e. $\large (\frac{a + \bar{a}}{2}) (\frac{z + \bar{z}}{2}) – (\frac{a – \bar{a}}{2i}) (\frac{z – \bar{z}}{2i})= 0 $

So, $\large \bar{\alpha} = a $ and $\large \alpha = \bar{a}$

Hence (B) and (C) are the correct answers.