Question

If $z$ be a complex number, then the minimum value of $\left| z-7 \right|+\left| z \right|$ is
(a) $\dfrac{\sqrt{3}-1}{2}$
(b) $-\sqrt{7}$
(c) $\dfrac{7+\sqrt{2}}{2}$
(d) $7$

Answer 1

Hint: Use the property of modulus of complex number,\[\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\ge \left| {{z}_{1}}+{{z}_{2}} \right|\ge \left| {{z}_{1}}-{{z}_{2}} \right|\], assume \[{{z}_{1}}=z-7\] and ${{z}_{2}}=z$. This property of modulus of complex numbers is also called triangle inequality property.

Complete answer:

A complex number is a number that can be expressed in the form of $a+ib$, where $a$ and $b$ are real numbers and $i$ is the solution of the equation ${{x}^{2}}=-1$. Because no real number satisfies this equation, $i$is called an imaginary number. For the complex number, $a+ib$, a is called the real part, and b is called the imaginary part. Complex numbers allow solutions to certain equations that have no solutions in real numbers. For any complex number, $z=x+iy$, the absolute value or modulus of $z$ is denoted $\left| z \right|$ and is defined by: \[\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}\], where $x\text{ and }y$ are the real and imaginary part of $z$ respectively. When the imaginary part $y$ is zero, this coincides with the definition of the absolute value of the real number $x$.

Now, we come to the question. We have to find the minimum value of complex function given by $\left| z-7 \right|+\left| z \right|$. We know that using triangle inequality we have,\[\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\ge \left| {{z}_{1}}+{{z}_{2}} \right|\ge \left| {{z}_{1}}-{{z}_{2}} \right|\], so, minimum value of the function $\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|$ is $\left| {{z}_{1}}-{{z}_{2}} \right|$.

Therefore, minimum of $\left| z-7 \right|+\left| z \right|$$=\left| (z-7)-z \right|=\left| -7 \right|=7$. Hence, option (d) is the correct answer.

Note: Since modulus of any number is always positive so, $\left| -7 \right|=7$. Don’t get confused that if inside modulus there is a negative number then if we remove modulus the number remains negative. It will always be positive. In other words, we can also say that minima occurs when $z$ becomes zero.