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Q.

$If the equation a | z |^{2} + \bar{\bar{α} z + α \bar{z}} + d = 0 represents a circle, where a, d are real constants,$ $T h e n w h i c h o f t h e f o l l o w i n g i s t r u e$

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a

$| α |^{2} - a d > 0 and a \in R - {0}$

b

$| α |^{2} - a d \geq 0 and a \in R$

c

$α = 0, a, d \in R^{+} ∣$

d

$| α |^{2} - a d \neq 0$

answer is A.

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Detailed Solution

$\begin{array}{l} a | z |^{2} + α \bar{z} + \bar{α} z + d = 0 \\ | z |^{2} + \frac{α}{a} \bar{z} + \frac{\bar{α}}{a} z + \frac{d}{a} = 0 \\ Centre = - \frac{α}{a} \\ r = \sqrt{{|\frac{α}{a}|}^{2} - \frac{d}{a}} ∣ \\ \frac{| α |^{2}}{a^{2}} - \frac{d}{a} > 0 | α |^{2} - a d > 0 & a \in R - {0} \end{array}$

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If the equation a|z|2+α¯z+αz¯¯+d=0 represents a circle, where a, d are real constants, Then which of the following is true