# Prove that ##12≤OP≤13## in the problem involving complex numbers

• chwala
In summary, the conversation discusses finding the minimum and maximum values of OP in a given equation, with the help of a diagram. The minimum value is found to be 12 and the maximum value is 13. The conversation also shows how by adding two equations, we can arrive at an inequality that helps us determine the range of OP, which is found to be between 12 and 13.
chwala
Gold Member
Homework Statement
See attached.
Relevant Equations
Complex numbers
Find the question below; note that no solution is provided for this question.

My approach;
Find part of my sketch here;

* My diagram may not be accurate..i just noted that, ##OP## takes smallest value of ##12## when ##|z+5|=|z-5|## i.e at the end of its minor axis and greatest value ##13## at end of its major axis

We have been given,
##|z-5|+(z+5|=26##
Then ##OP=|z|=|\dfrac{1}{2}((z+5)+(z-5))|##
##≤ \dfrac{1}{2}(|z+5|+|z-5|)=\dfrac{1}{2}×26=13##

Also,
##(|z+5|+|z-5|)^2=676## and ##(|z+5|-|z-5|)^2≥0##, adding this two gives us
##2|z+5|^2+2|z-5|^2≥676##
##⇒|z+5|^2+|z-5|^2≥338##
##(z+5)(z^*+5^*)+(z-5)(z^*-5^*)≥338##
##2zz^*+50≥338##
##2zz^*≥288##
##zz^*≥144##, which is ##|z|^2≥144##
##⇒z≥12##, therefore ##12≤OP≤13##

Last edited:
PeroK
Looks good.

## 1. What does OP stand for in the problem involving complex numbers?

OP stands for the distance between the origin (0,0) and a point P on the complex plane. It is also known as the modulus or absolute value of the complex number.

## 2. How do you prove that 12 ≤ OP ≤ 13?

To prove that 12 ≤ OP ≤ 13, we need to show that the distance between the origin and any point P on the complex plane falls within the range of 12 to 13. This can be done by using the Pythagorean theorem and the properties of complex numbers.

## 3. Why is it important to prove that 12 ≤ OP ≤ 13 in the problem involving complex numbers?

Proving that 12 ≤ OP ≤ 13 is important because it helps us understand the properties of complex numbers and their geometric representation on the complex plane. It also allows us to solve problems involving complex numbers more accurately.

## 4. Can you provide an example of a complex number that satisfies 12 ≤ OP ≤ 13?

Yes, an example of a complex number that satisfies 12 ≤ OP ≤ 13 is 6 + 8i. The distance between the origin and this point P is √(6^2 + 8^2) = 10, which falls within the given range.

## 5. Are there any other methods to prove that 12 ≤ OP ≤ 13 besides using the Pythagorean theorem?

Yes, there are other methods such as using the properties of complex numbers, using the distance formula, or using the triangle inequality. However, the Pythagorean theorem is the most commonly used method in proving the range of OP in the problem involving complex numbers.

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