Which Complex Number Has A Distance Of 17 \sqrt{17} 17 ​ From The Origin On The Complex Plane?A. 2 + 15 I 2 + 15i 2 + 15 I B. 17 + I 17 + I 17 + I C. 20 − 3 I 20 - 3i 20 − 3 I D. 4 − I 4 - I 4 − I

Introduction

In the complex plane, the distance of a complex number from the origin is given by the modulus or absolute value of the complex number. This distance is calculated using the formula $|a + bi| = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number, respectively. In this article, we will explore which complex number has a distance of $\sqrt{17}$ from the origin on the complex plane.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part of a complex number is denoted by $a$, and the imaginary part is denoted by $b$. The modulus or absolute value of a complex number is denoted by $|a + bi|$.

Calculating the Distance of a Complex Number from the Origin

The distance of a complex number from the origin is given by the formula $|a + bi| = \sqrt{a^2 + b^2}$. This formula can be used to calculate the distance of any complex number from the origin.

Analyzing the Options

We are given four complex numbers: $2 + 15i$, $17 + i$, $20 - 3i$, and $4 - i$. We need to calculate the distance of each of these complex numbers from the origin and determine which one has a distance of $\sqrt{17}$.

Option A: $2 + 15i$

To calculate the distance of $2 + 15i$ from the origin, we use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Substituting $a = 2$ and $b = 15$, we get:

$$|2 + 15i| = \sqrt{2^2 + 15^2} = \sqrt{4 + 225} = \sqrt{229}$$

The distance of $2 + 15i$ from the origin is $\sqrt{229}$, which is not equal to $\sqrt{17}$.

Option B: $17 + i$

To calculate the distance of $17 + i$ from the origin, we use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Substituting $a = 17$ and $b = 1$, we get:

$$|17 + i| = \sqrt{17^2 + 1^2} = \sqrt{289 + 1} = \sqrt{290}$$

The distance of $17 + i$ from the origin is $\sqrt{290}$, which is not equal to $\sqrt{17}$.

Option C: $20 - 3i$

To calculate the distance of $20 - 3i$ from the origin, we use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Substituting $a = 20$ and $b = -3$, we get:

$$|20 - 3i| = \sqrt{20^2 + (-3)^2} = \sqrt{400 + 9} = \sqrt{409}$$

The distance of $20 - 3i$ from the origin is $\sqrt{409}$, which is not equal to $\sqrt{17}$.

Option D: $4 - i$

To calculate the distance of $4 - i$ from the origin, we use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Substituting $a = 4$ and $b = -1$, we get:

$$|4 - i| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}$$

The distance of $4 - i$ from the origin is $\sqrt{17}$, which is equal to the given distance.

Conclusion

Based on the calculations, we can conclude that the complex number $4 - i$ has a distance of $\sqrt{17}$ from the origin on the complex plane.

Final Answer

The final answer is $\boxed{D}$

Introduction

In our previous article, we explored which complex number has a distance of $\sqrt{17}$ from the origin on the complex plane. We analyzed four complex numbers and determined that the complex number $4 - i$ has a distance of $\sqrt{17}$ from the origin. In this article, we will provide a Q&A section to further clarify the concept and answer any additional questions.

Q&A

Q: What is the distance of a complex number from the origin?

A: The distance of a complex number from the origin is given by the modulus or absolute value of the complex number, which is calculated using the formula $|a + bi| = \sqrt{a^2 + b^2}$.

Q: How do I calculate the distance of a complex number from the origin?

A: To calculate the distance of a complex number from the origin, you can use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Simply substitute the real and imaginary parts of the complex number into the formula and calculate the result.

Q: What is the difference between the modulus and the absolute value of a complex number?

A: The modulus and the absolute value of a complex number are the same thing. They both refer to the distance of the complex number from the origin.

Q: Can I use the distance formula to calculate the distance of a complex number from a point other than the origin?

A: No, the distance formula is only used to calculate the distance of a complex number from the origin. If you want to calculate the distance of a complex number from a point other than the origin, you will need to use a different formula.

Q: How do I determine which complex number has a certain distance from the origin?

A: To determine which complex number has a certain distance from the origin, you can use the formula $|a + bi| = \sqrt{a^2 + b^2}$. Simply substitute the given distance into the formula and solve for the real and imaginary parts of the complex number.

Q: Can I use the distance formula to calculate the distance of a complex number from the origin if the complex number is in polar form?

A: Yes, you can use the distance formula to calculate the distance of a complex number from the origin if the complex number is in polar form. The formula is $|r(\cos \theta + i \sin \theta)| = r$, where $r$ is the magnitude of the complex number and $\theta$ is the argument of the complex number.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the formulas $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1} \left(\frac{b}{a}\right)$, where $a$ and $b$ are the real and imaginary parts of the complex number, respectively.

Conclusion

We hope this Q&A section has helped to clarify any questions you may have had about the distance of a complex number from the origin. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is $\boxed{D}$