Let z1 - 18 - 83i- z2 - 18 - 39i- ana z3- 78 - 99i- where i - -1- Let z be a unique comlpex number with the properties that z3 - z1z2 - z1 - z - z2z - z3 is a real number and the imaginary part of the size z is the greatest possible-

The correct option is A Re (z) = 56 Using the property that #160;if z_3 z_1z_2 z_1.z z_2z z_3 is a real number then the four points are concyclic i. ...

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Byju's Answer

Standard XII

Mathematics

Complex Numbers

Let z1 = 18...

Question

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 $\frac{z_{3} - z_{1}}{z_{2} - z_{1}}$ $\cdot$ $\frac{z - z_{2}}{z - z_{3}}$ is a real number and the imaginary part of the size z is the greatest possible.

R e (z) = 56

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R e (z) = 61

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R e (z) = 54

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R e (z) = 59

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Solution

The correct option is A $R e (z) = 56$
Using the property that

if $\frac{z_{3} - z_{1}}{z_{2} - z_{1}} . \frac{z - z_{2}}{z - z_{3}}$ 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 $z_{1} z_{2}$
$y = 61$

Equation of perpendicular bisector of $z_{3} z_{2}$
$y + x = 117$

Solving the equation we get,
$x = 56$

So, $R e (z) = 56$

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Let z1 = 18 + 83i, z2 = 18 + 39i, ana z3= 78 + 99i. where i = √−1. Let z be a unique comlpex number with the properties that z3−z1z2−z1 ⋅ z−z2z−z3 is a real number and the imaginary part of the size z is the greatest possible.