Find Complex Numbers: Distance $\sqrt{17}$ From Origin

Hey guys! Ever wondered how to pinpoint specific numbers on the complex plane? It's kinda like navigating a map, but instead of streets, we've got real and imaginary axes. Today, we're tackling a super cool problem: finding a complex number that's a specific distance away from the origin. Our target distance? A neat $\sqrt{17}$. This isn't just about memorizing formulas; it's about understanding the geometry behind complex numbers. We'll break down why the distance from the origin is so important and how it relates to the magnitude of a complex number. Get ready to flex those math muscles because we're diving deep into the fascinating world of complex numbers and their distances. We've got some options to choose from, and by the end of this, you'll be able to confidently identify the right one. Let's get started and make complex numbers less intimidating and more awesome!

Understanding the Complex Plane and Distance

So, what exactly are we talking about when we say the "complex plane"? Imagine a regular graph, but instead of just an x-axis and a y-axis, we have a real axis (like the x-axis) and an imaginary axis (like the y-axis). A complex number, typically written as $a + bi$, is represented as a point $(a, b)$ on this plane. Here, '$a$' is the real part, and '$b$' is the imaginary part. The 'i' is the imaginary unit, you know, the square root of -1. Now, when we talk about the distance from the origin on this complex plane, we're essentially talking about how far that point $(a, b)$ is from the point $(0, 0)$. This distance is super important because it's also known as the magnitude or modulus of the complex number. It tells us the 'size' of the complex number. To calculate this distance, we use the Pythagorean theorem. If our complex number is $z = a + bi$, its distance from the origin, denoted as $|z|$, is calculated as $\sqrt{a^2 + b^2}$. This formula comes straight from the Pythagorean theorem, where '$a$' is one leg of a right triangle, '$b$' is the other leg, and the distance $|z|$ is the hypotenuse. So, for our problem, we're looking for a complex number $a + bi$ such that $\sqrt{a^2 + b^2} = \sqrt{17}$. This means we need to find values for '$a$' and '$b$' where $a^2 + b^2 = 17$. This is the core equation we'll be working with to solve the problem. Understanding this fundamental concept is key to unlocking the solution, so let's make sure this part is crystal clear, guys!

Evaluating the Options: The Math Behind Each Choice

Alright, team, we've got our mission: find the complex number whose magnitude (distance from the origin) is $\sqrt{17}$. This means we need to find the pair $(a, b)$ from the options where $a^2 + b^2 = 17$. Let's put on our detective hats and analyze each option one by one. This is where the real action happens, and we'll see which one fits the bill. Remember, the formula is $|z| = \sqrt{a^2 + b^2}$, so we're really checking if $a^2 + b^2 = 17$ for each choice.

Option A: $2 + 15i$

First up is $2 + 15i$. Here, our real part '$a$' is 2, and our imaginary part '$b$' is 15. Let's plug these into our equation: $a^2 + b^2$. So, we have $2^2 + 15^2$. Calculate that: $2^2$ is 4, and $15^2$ is a whopping 225. Adding them together gives us $4 + 225 = 229$. Is 229 equal to 17? Nope, not even close! So, option A is definitely not our winner. It's way too far from the origin.

Option B: $17 + i$

Next, we have option B: $17 + i$. For this complex number, '$a$' is 17 and '$b$' is 1. Let's see what $a^2 + b^2$ gives us: $17^2 + 1^2$. Okay, $17^2$ is 289, and $1^2$ is just 1. Adding them up, we get $289 + 1 = 290$. Again, 290 is nowhere near 17. So, option B is also out of the running. This one is even further out than option A, which is pretty wild!

Option C: $20 - 3i$

Moving on to option C: $20 - 3i$. Here, '$a$' is 20, and '$b$' is -3. Remember, when we square '$b$', the negative sign disappears. So, let's calculate: $a^2 + b^2 = 20^2 + (-3)^2$. $20^2$ is 400, and $(-3)^2$ is 9. Adding these gives us $400 + 9 = 409$. Is 409 equal to 17? Absolutely not. Option C is also a bust. Looks like we're getting some pretty big numbers here, which means these complex numbers are quite far from the origin.

Option D: $4 - i$

Finally, we arrive at option D: $4 - i$. For this complex number, '$a$' is 4, and '$b$' is -1. Let's calculate $a^2 + b^2$: $4^2 + (-1)^2$. $4^2$ is 16, and $(-1)^2$ is 1. Adding these gives us $16 + 1 = 17$. Bingo! We found it! $16 + 1$ is exactly 17. This means the distance from the origin for $4 - i$ is $\sqrt{17}$. So, option D is our correct answer, guys! It’s awesome when the math just clicks, right?

Confirming the Solution and Key Takeaways

We've successfully navigated through all the options, and option D, $4 - i$, is the complex number that has a distance of $\sqrt{17}$ from the origin. We confirmed this by calculating its magnitude: $|4 - i| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}$. It’s super satisfying when you find the answer and know exactly why it's correct. This problem really highlights the geometric interpretation of complex numbers. The magnitude, or distance from the origin, is a fundamental property that tells us about the 'size' or 'strength' of a complex number. Remember, for any complex number $z = a + bi$, its distance from the origin is always calculated using the Pythagorean theorem: $|z| = \sqrt{a^2 + b^2}$. This means we're looking for $a^2 + b^2$ to equal the square of the desired distance. In our case, the desired distance was $\sqrt{17}$, so we needed $a^2 + b^2 = 17$. It’s also worth noting that the other options resulted in much larger distances. For instance, option A ($2+15i$) had a distance of $\sqrt{229}$, option B ($17+i$) had a distance of $\sqrt{290}$, and option C ($20-3i$) had a distance of $\sqrt{409}$. These values are significantly larger than $\sqrt{17}$, confirming that they are indeed incorrect. Understanding this concept not only helps solve problems like this but also is crucial for more advanced topics in complex analysis, like plotting circles centered at the origin (where all points have the same distance from the origin) or understanding the behavior of functions. So, next time you see a complex number, think about where it sits on the plane and how far it is from that central point. It's a powerful way to visualize and understand these numbers. Keep practicing, guys, and you'll master this in no time!

Final Answer

The complex number that has a distance of $\sqrt{17}$ from the origin on the complex plane is D. $4-i$. We reached this conclusion by applying the distance formula for complex numbers, which is derived from the Pythagorean theorem, to each of the given options. The formula states that the distance (or magnitude) of a complex number $a + bi$ from the origin is $|a + bi| = \sqrt{a^2 + b^2}$. We needed to find the option where $\sqrt{a^2 + b^2} = \sqrt{17}$, which simplifies to finding the option where $a^2 + b^2 = 17$. Evaluating each option:

• A. $2+15i$: $2^2 + 15^2 = 4 + 225 = 229 eq 17$
• B. $17+i$: $17^2 + 1^2 = 289 + 1 = 290 eq 17$
• C. $20-3i$: $20^2 + (-3)^2 = 400 + 9 = 409 eq 17$
• D. $4-i$: $4^2 + (-1)^2 = 16 + 1 = 17$. This matches our requirement. Therefore, $4-i$ is the correct answer. This exercise reinforces the connection between algebra and geometry in the realm of complex numbers, showing how simple algebraic calculations can reveal spatial relationships on the complex plane. Keep exploring, and happy calculating!