Question:
The value of $|z|^2 + |z - 3|^2 + |z - i|^2$ is minimum when $z$ equals.
The value of $|z|^2 + |z - 3|^2 + |z - i|^2$ is minimum when $z$ equals.
Updated On: Jul 7, 2022
- $2 \,\frac {2} {3} i$
- $45 + 3i$
- $1+ \frac {i} {3} $
- $1 \,\frac {i} {3} $
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The Correct Option is C
Solution and Explanation
$\left|z\right|^{2}+\left|z-3\right|^{2}+\left|z-1\right|^{2}$
$=x^{2}+y^{2}+\left(x-3\right)^{2}+y^{2}+x^{2}\left(y-1\right)^{2}$
$=3x^{2}+3y^{2}-6x-2y+10$
$=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2y}{3}\right)+10-3$
$=3\left(x-1\right)^{2}+3\left(y^{2}-\frac{2}{3}\right)^{2}+7-3$
$=3\left|z-\left(1+\frac{i}{3}\right)\right|^{2}+\frac{20}{3}$
This is minimum if $z=1+\frac{i}{3}$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
