Complex Disguise!

Algebra Level 4

Complex numbers $z_1$ , $z_2$ and $z_3$ have the following properties.

$|z_1|=|z_2|=|z_3|=5$
$z_1+z_2=0$

What is the value of $(z_3-z_1)(\overline{z_3-z_1})+(z_3-z_2)(\overline{z_3-z_2})$ ?

The answer is 100.

1 solution

Anish Puthuraya
Jan 22, 2014

Consider a Circle of Radius $5$ in the Argand Plane.
From the First Information,
It is clear that all the 3 complex numbers lie on this circle.

From the Second Information,
we conclude that $z_1$ and $z_2$ are 2 opposite vectors.
Let both of these vectors lie on the real axis (as it does not affect the answer; just for simplicity)

Drawing a clear diagram, It is evident that $z_1z_2$ forms a diameter of the circle, and
$z_3$ is a random point on the circle.

Now,
By the properties of Circles, we have,
$(z_3-z_1)$ is Perpendicular to $(z_3-z_2)$ .

Also,
$z\bar{z}=|z|^2$

Thus, the problem simplifies to,
$|(z_3-z_1)|^2+|(z_3-z_2)|^2$

Applying Pythogoras Theorem,
we get, $|(z_3-z_1)|^2+|(z_3-z_2)|^2 = (2r)^2 = (10)^2 = \boxed{100}$

Though I didn't write a solution, but there is a better solution:

We know that:

$|z_{3} - z_{1}|^2 + |z_{3}-z_{2}|^2$

= $|z_{1}|^2 + |z_{3}|^2 + |z_{3}|^2 + |z_{2}|^2 + 2Re(z_{3} \bar{z_{2}}) + 2Re(z_{3} \bar{z_{1}})$

= $100 + 2Re (z_{3} (\overline{z_{1} + z_{2}})) = 100$ as $z_{1} + z_{2} = 0$

jatin yadav - 7 years, 4 months ago

Complex Disguise!

The answer is 100.

1 solution

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