Let $z_1$ = 18 + 83i, $z_2$ = 18 + 39i, ana $z_3$= 78 + 99i. where i = $\sqrt-1$. Let z be a unique comlpex number with the properties that $\dfrac{z_3 - z_1}{z_2 - z_1}$ $\cdot$ $\dfrac{z - z_2}{z - z_3}$ is a real number and the imaginary part of the size z is the greatest possible.
• A
$Re (z) = 56$
• A
$Re (z) = 56$
• B
$Re (z) = 61$
• B
$Re (z) = 61$
• C
$Re (z) = 54$
• C
$Re (z) = 54$
• D
$Re (z) = 59$
• D
$Re (z) = 59$

#### Solution

Using the property that

if $\dfrac{z_3-z_1}{z_2-z_1}.\dfrac{z-z_2}{z-z_3}$is a real number then the four points are concyclic i.e. are present on a circle

since it is told the imaginary part is maximum so,it the top point of the circle
so we take the perpendicular bisector of any two line and take its intersection.
As shown in the figure,

Equation of perpendicular bisector of $z_1z_2$
$y=61$

Equation of perpendicular bisector of $z_3z_2$
$y+x=117$

Solving the equation we get,
$x=56$

So, $Re(z)=56$