Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y ( x , y , ? i = - 1 ) of the equation s z + t z + r = 0 , where z = x - i y . Then, which of the following statement(s) is (are) TRUE [20

Question

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y$ $(x, y \in ℝ, i = \sqrt{- 1})$ of the equation

$s z + t \bar{z} + r = 0, where \bar{z} = x - i y .$ Then, which of the following statement(s) is (are) TRUE? [2018]

Smart Achievers · Accepted Answer

(1, 3, 4)

$We have,$

$s z + t \bar{z} + r = 0 \dots (i)$

$On taking conjugate$

$\bar{s} \bar{z} + \bar{t z} + \bar{r} = 0 \dots (i i)$

$On solving Eqs. (i) and (ii), we get$

$z = \frac{\bar{r} t - r \bar{s}}{| s |^{2} - | t |^{2}}$

$(a) For unique solutions of z$

$| s |^{2} - | t |^{2} \neq 0 \Rightarrow | s | \neq | t |$

$It is true.$

$(b) If | s | = | t |, then \bar{r} t - r \bar{s} may or may not be zero.$

$So, z may have no solution.$

$∴$ $L may be an empty set.$

$It is false.$

$(c)$ If elements of set L represents line, then this line and given circle intersect at maximum two points.

Hence, it is true.

$(d)$ In the case locus of $z$ is a line, so L has infinite elements.

Hence, it is true.

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