Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y \in R, i = \sqrt{- 1}) of he equation \text{sz}\textrm{ } + t \bar{z} + \textrm{ }\text{r}\textrm{ } = \textrm{ } 0 \textrm{ } where \textrm{ } \bar{z} = \textrm{ }\text{x}\textrm{ } – \textrm{ }\text{iy}. Then which of the following statement(s) is(are) TRUE?

Engineering

Mathematics

Geometry of Complex Number

Question

Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy (x, y $\in$ R, i = $\sqrt{- 1}$ ) of he equation $sz + t \bar{z} + r = 0 where \bar{z} = x - iy$ . Then which of the following statement(s) is(are) TRUE?

If L has more than one element, then L has infinitely many elements

If L has exactly one element, then | s | $\neq$ | t |

The number of elements in L $\cap$ {z : | z – 1 + i | = 5} is at most 2

If | s | = | t |, then L has infinitely many elements

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$sz + t \bar{z} + r = 0$

$\bar{s} \bar{z} + \bar{t} z + \bar{r} = 0$

$\Rightarrow s \bar{s} z + \bar{s} t \bar{z} + \bar{s} r = 0$

$t \bar{s} \bar{z} + t \bar{t} z + t \bar{r} = 0$

$z ({| s |}^{2} - {| t |}^{2}) + \bar{s} r - t \bar{r} = 0$

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