Finding The Focus Of A Parabola X² = -20y A Step By Step Guide

Hey math enthusiasts! Today, we're diving deep into the fascinating world of parabolas, specifically focusing on how to pinpoint the coordinates of the focus when given the equation. We'll be tackling the equation x² = -20y and breaking down the steps to identify its focus. So, grab your thinking caps, and let's get started!

Understanding the Parabola Equation

In this section, we will be discussing the key parabola equation, which is crucial for solving this problem and others like it. The equation we're working with, x² = -20y, is a classic representation of a parabola. But what does it really tell us? First, it's important to understand the standard forms of parabola equations. When we talk about parabolas opening upwards or downwards, the standard form is x² = 4py. On the flip side, for parabolas opening to the right or left, we use the form y² = 4px. The variable 'p' here is super important because it dictates the distance between the vertex (the turning point of the parabola) and the focus, as well as the distance between the vertex and the directrix (a line that helps define the parabola's shape). Understanding these standard forms is crucial because it allows us to quickly identify key characteristics of the parabola, such as its orientation (whether it opens up, down, left, or right) and the distance between the vertex and the focus. In our case, we have x² = -20y, which closely resembles the standard form x² = 4py. This tells us immediately that our parabola opens either upwards or downwards. The negative sign in front of the 20 is the key here. It indicates that the parabola actually opens downwards. Think of it like this: the negative sign 'flips' the parabola upside down compared to a standard x² = 4py equation where 'p' is positive. Now, let's dig deeper into how to extract information from this equation. To find the focus, we need to figure out the value of 'p'. This value will tell us how far the focus is from the vertex. To do this, we'll compare our equation, x² = -20y, with the standard form x² = 4py. By equating the coefficients, we can solve for 'p'. This is a critical step in finding the focus, so pay close attention! Once we have 'p', we'll know the distance between the vertex and the focus. Remember, the vertex is the point where the parabola changes direction, and the focus is a special point inside the curve that helps define its shape. In the next section, we'll walk through the steps of actually solving for 'p' and then using that value to find the focus coordinates. So, keep those thinking caps on, and let's move forward!

Solving for 'p'

Now, guys, let's get down to the nitty-gritty and solve for 'p'! This is a crucial step in finding the coordinates of the focus. Remember our equation: x² = -20y. And remember the standard form for a parabola opening upwards or downwards: x² = 4py. Our goal here is to match these two equations up so we can figure out what 'p' is. It's like solving a puzzle, and 'p' is the missing piece! To do this, we need to equate the coefficients of 'y' in both equations. In our equation, x² = -20y, the coefficient of 'y' is -20. In the standard form, x² = 4py, the coefficient of 'y' is 4p. So, we can set up a simple equation: 4p = -20. See how we're taking the part of the equation that multiplies the 'y' and setting them equal to each other? This is the key to unlocking the value of 'p'. Now, to isolate 'p', we need to get rid of the 4 that's multiplying it. We can do this by dividing both sides of the equation by 4. This is a fundamental algebraic operation that keeps the equation balanced. When we divide both sides of 4p = -20 by 4, we get p = -20 / 4. Time for some simple arithmetic! -20 divided by 4 is -5. So, we've found our 'p'! p = -5. This is a crucial result, so make sure you understand how we got here. We matched the coefficients, set up an equation, and solved for 'p'. Now that we know 'p', we're one giant step closer to finding the focus. But what does p = -5 actually mean? Remember, 'p' represents the distance between the vertex and the focus. The negative sign is also super important. It tells us that the parabola opens downwards (we already suspected this because of the negative sign in the original equation, but now we have confirmation!). If 'p' were positive, the parabola would open upwards. The magnitude of 'p', which is 5 in this case, tells us the actual distance. So, the focus is 5 units away from the vertex. But in which direction? Since the parabola opens downwards, the focus will be located below the vertex. In the next section, we'll use this information to determine the exact coordinates of the focus. We'll put all the pieces together – the value of 'p', the direction the parabola opens, and the vertex location – to pinpoint the focus. So, stick with me, and let's finish this puzzle!

Finding the Focus Coordinates

Alright, mathletes, we've cracked the code for 'p'! Now comes the moment we've been building up to: finding the focus coordinates. This is where all our hard work pays off, and we get to see the practical application of our knowledge. Remember, we found that p = -5. This tells us the distance between the vertex and the focus, and the negative sign tells us the parabola opens downwards. But where exactly is the focus located? To figure this out, we need to consider the vertex of the parabola. The vertex is the turning point of the parabola, the point where it changes direction. For parabolas in the form x² = 4py or y² = 4px, when there are no additional terms added or subtracted from x or y, the vertex is conveniently located at the origin, which is the point (0, 0). This makes our lives much easier! So, we know our vertex is at (0, 0), and we know the focus is 5 units away in the downward direction (because p = -5). This is like having a starting point on a map (the vertex) and a direction and distance to travel (downwards, 5 units) to find our treasure (the focus). Since we're moving downwards from the origin, we're changing the y-coordinate. The x-coordinate will stay the same because we're not moving left or right. To find the y-coordinate of the focus, we need to subtract the magnitude of 'p' (which is 5) from the y-coordinate of the vertex (which is 0). This is because we're moving downwards, which means decreasing the y-value. So, the y-coordinate of the focus is 0 - 5 = -5. And since the x-coordinate remains at 0, the coordinates of the focus are (0, -5). Boom! We found it! We've successfully located the focus of the parabola x² = -20y. This is a fantastic achievement, and you should be proud of yourself for following along. Let's recap what we did: We identified the standard form of the equation, solved for 'p', determined the direction the parabola opens, identified the vertex, and then used the value of 'p' and the vertex location to find the focus coordinates. In the next section, we'll look at the answer choices provided in the question and see which one matches our result. We'll also briefly discuss why the other options are incorrect. This will solidify our understanding and help us avoid common mistakes.

Checking the Answer Choices

Okay, team, we've pinpointed the focus coordinates as (0, -5). Now, let's put our detective hats on and see which of the answer choices matches our findings. This is like the final puzzle piece snapping into place, confirming we've solved the mystery correctly! The answer choices provided were:

A. (-5, 0) B. (5, 0) C. (0, 5) D. (0, -5)

Looking at these options, it's clear that D. (0, -5) perfectly aligns with the focus coordinates we calculated. Hooray! We've successfully navigated the problem and arrived at the correct answer. But before we celebrate too much, let's briefly analyze why the other options are incorrect. This is a valuable exercise because it helps us understand the underlying concepts more deeply and avoid similar mistakes in the future. Option A, (-5, 0), has the correct numbers but the coordinates are flipped. This would represent a point 5 units to the left of the origin, which is not the focus of our parabola. It's important to remember that the order of coordinates matters: (x, y) is not the same as (y, x). Option B, (5, 0), is similar to option A but represents a point 5 units to the right of the origin. Again, this is not the focus of our parabola. Option C, (0, 5), has the correct x-coordinate but the wrong y-coordinate. This point is 5 units above the origin. While it's in the correct vertical line, it's in the opposite direction from where the focus should be (remember, our parabola opens downwards). By understanding why these options are incorrect, we reinforce our understanding of parabolas and how the focus is determined by the equation. We've not only found the right answer but also deepened our knowledge in the process! In conclusion, we've successfully determined the focus of the parabola x² = -20y by understanding the standard form of the equation, solving for 'p', identifying the vertex, and using this information to calculate the focus coordinates. We then confirmed our answer by checking the provided answer choices and ruling out the incorrect options. You guys have done an amazing job following along! Keep practicing, and you'll become parabola masters in no time! Remember, math is like a muscle; the more you use it, the stronger it gets. So, keep those brains flexing and keep exploring the fascinating world of mathematics!

Final Answer: The final answer is (D)

This is the final part of the article and contains only the final answer.