Finding The Focus Of A Parabola X^2=4y A Comprehensive Guide

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Introduction to Parabolas

In the fascinating world of mathematics, parabolas stand out as essential curves with a multitude of applications. From the trajectory of a ball thrown in the air to the design of satellite dishes, parabolas play a crucial role. Understanding the properties of a parabola, especially its focus, is fundamental in various fields of science and engineering. In this comprehensive guide, we will delve into the specifics of parabolas, focusing on how to determine the coordinates of the focus given the equation of the parabola.

This article aims to provide a thorough explanation of parabolas, their equations, and the method to find the focus. We will address the question: "A parabola is represented by the equation x2=4yx^2=4 y. What are the coordinates of the focus of the parabola?" by breaking down the problem step by step. We will explore the standard form of a parabola, the significance of its parameters, and the practical application of these concepts. Whether you are a student learning about conic sections or someone looking to refresh your knowledge, this guide will provide a clear and detailed explanation.

What is a Parabola?

A parabola is a U-shaped curve that can be defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This definition is crucial for understanding the geometric properties of a parabola. The line passing through the focus and perpendicular to the directrix is the axis of symmetry, and the point where the parabola intersects its axis of symmetry is the vertex. Understanding these basic elements is key to solving problems related to parabolas.

Standard Equation of a Parabola

To determine the coordinates of the focus, it's essential to understand the standard equations of a parabola. A parabola can open upwards, downwards, leftwards, or rightwards, each corresponding to a different standard form. For a parabola that opens upwards or downwards, the standard equation is given by:

(x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)

where:

  • (h, k) represents the coordinates of the vertex.
  • p is the distance from the vertex to the focus and from the vertex to the directrix.

The sign of p determines the direction in which the parabola opens. If p > 0, the parabola opens upwards, and if p < 0, it opens downwards.

For a parabola that opens leftwards or rightwards, the standard equation is given by:

(y−k)2=4p(x−h)(y - k)^2 = 4p(x - h)

In this case, if p > 0, the parabola opens rightwards, and if p < 0, it opens leftwards.

The given equation in the problem is x2=4yx^2 = 4y, which is a special case of the standard form where h = 0 and k = 0. This simplifies the equation and helps us to easily identify the key parameters.

Analyzing the Given Equation: x2=4yx^2 = 4y

The equation provided, x2=4yx^2 = 4y, is a specific form of a parabola that opens either upwards or downwards. By comparing this equation to the standard form (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k), we can identify the vertex and the value of p. This is a critical step in finding the coordinates of the focus.

Identifying the Vertex

The vertex of a parabola is the point at which the curve changes direction. In the standard equation (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k), the vertex is given by the coordinates (h, k). For the equation x2=4yx^2 = 4y, we can rewrite it as (x−0)2=4(1)(y−0)(x - 0)^2 = 4(1)(y - 0). This form clearly shows that:

  • h = 0
  • k = 0

Therefore, the vertex of the parabola is at the origin, (0, 0). The vertex serves as a reference point from which we can determine the location of the focus and directrix.

Determining the Value of p

The parameter p represents the distance from the vertex to the focus and from the vertex to the directrix. In the equation x2=4yx^2 = 4y, we can see that the equation is in the form (x−0)2=4p(y−0)(x - 0)^2 = 4p(y - 0). Comparing this to the given equation x2=4yx^2 = 4y, we can deduce that:

4p=44p = 4

Dividing both sides by 4, we find:

p=1p = 1

The value of p is crucial because it not only gives us the distance but also the direction in which the parabola opens. Since p = 1, which is positive, the parabola opens upwards. This means the focus will be located above the vertex, and the directrix will be a horizontal line below the vertex.

Finding the Focus of the Parabola

Now that we have identified the vertex (0, 0) and the value of p (1), we can determine the coordinates of the focus. For a parabola that opens upwards, the focus is located at a distance of p units above the vertex. The formula for the focus of a parabola opening upwards is:

Focus: (h, k + p)

Applying the Formula

Using the values we found:

  • h = 0
  • k = 0
  • p = 1

We can substitute these into the formula for the focus:

Focus: (0, 0 + 1)

Therefore, the coordinates of the focus are:

Focus: (0, 1)

This result tells us that the focus of the parabola x2=4yx^2 = 4y is located at the point (0, 1) on the Cartesian plane. This point is essential for understanding the properties and applications of the parabola.

Detailed Explanation of the Solution

To recap, we started with the equation of the parabola x2=4yx^2 = 4y and systematically identified the coordinates of its focus. This process involved several key steps, each building upon the previous one to arrive at the final answer. Let's walk through these steps in detail to reinforce the concepts.

  1. Identify the Standard Form: The first step was recognizing that the given equation is a form of a parabola that opens either upwards or downwards. Comparing it to the standard form (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k) is essential for identifying the parameters.
  2. Determine the Vertex: By rewriting the equation x2=4yx^2 = 4y as (x−0)2=4(1)(y−0)(x - 0)^2 = 4(1)(y - 0), we identified the vertex coordinates as (h, k) = (0, 0). The vertex is a crucial reference point for locating other key features of the parabola.
  3. Find the Value of p: The parameter p represents the distance from the vertex to the focus and the vertex to the directrix. By equating 4p4p to 4 in the equation, we found that p=1p = 1. This value not only gives us the distance but also indicates the direction the parabola opens (upwards since p is positive).
  4. Apply the Focus Formula: For a parabola that opens upwards, the focus is located at (h, k + p). Substituting the values h = 0, k = 0, and p = 1 into this formula, we calculated the focus coordinates as (0, 0 + 1) = (0, 1).

Each of these steps is critical for a thorough understanding of how to find the focus of a parabola given its equation. The systematic approach ensures accuracy and provides a clear pathway to the solution.

Conclusion: Coordinates of the Focus

In conclusion, for the parabola represented by the equation x2=4yx^2 = 4y, the coordinates of the focus are (0, 1). This result was obtained by systematically analyzing the equation, identifying the vertex and the parameter p, and then applying the formula for the focus of an upward-opening parabola. This exercise highlights the importance of understanding the standard forms of parabolic equations and their parameters.

Final Answer

The correct answer is B. (0, 1).

This comprehensive guide has walked you through the process of finding the focus of a parabola, providing a solid foundation for tackling similar problems in mathematics and related fields. Understanding the properties of parabolas is not only crucial for academic success but also for practical applications in engineering and science.