Finding the Focus of a Parabola

Oct 13, 2025 by pynkocean.com 23 views

Introduction

In mathematics, a parabola is a type of quadratic curve that has a specific shape. It is defined as the set of all points that are equidistant to the focus and the directrix. The focus is a fixed point that is used to define the parabola, and the directrix is a fixed line that is perpendicular to the axis of symmetry of the parabola. In this article, we will explore the properties of a parabola with a vertex at the origin and a directrix at y = 3, and we will find the coordinates of its focus.

Properties of a Parabola

A parabola is a quadratic curve that has a specific shape. It is defined as the set of all points that are equidistant to the focus and the directrix. The focus is a fixed point that is used to define the parabola, and the directrix is a fixed line that is perpendicular to the axis of symmetry of the parabola. The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.

The equation of a parabola with a vertex at the origin and a directrix at y = 3 can be written in the form:

y = ax^2

where a is a constant that determines the shape of the parabola. The focus of a parabola is a fixed point that is used to define the parabola, and it is located at a distance of 1/4a from the vertex along the axis of symmetry.

Finding the Focus of a Parabola

To find the focus of a parabola, we need to know the value of a. Since the directrix is at y = 3, we can write the equation of the parabola as:

y = ax^2

We also know that the vertex is at the origin, so the equation of the axis of symmetry is x = 0. Since the directrix is at y = 3, the distance from the vertex to the directrix is 3.

Using the formula for the distance from the vertex to the focus, we can write:

distance = 1/4a

Since the distance from the vertex to the directrix is 3, we can set up the equation:

3 = 1/4a

Solving for a, we get:

a = -4/3

Now that we have the value of a, we can find the coordinates of the focus. The focus is located at a distance of 1/4a from the vertex along the axis of symmetry. Since the axis of symmetry is x = 0, the x-coordinate of the focus is 0. The y-coordinate of the focus is the distance from the vertex to the focus, which is 1/4a.

Substituting the value of a, we get:

y-coordinate = 1/4(-4/3) = -1/3

So, the coordinates of the focus are (0, -1/3).

Conclusion

In this article, we explored the properties of a parabola with a vertex at the origin and a directrix at y = 3. We found the equation of the parabola and used it to determine the value of a. We then used the formula for the distance from the vertex to the focus to find the coordinates of the focus. The coordinates of the focus are (0, -1/3).

Final Answer

The final answer is (0, -1/3).

Discussion

The discussion category for this article is mathematics. The article covers the properties of a parabola with a vertex at the origin and a directrix at y = 3, and it provides a step-by-step solution to find the coordinates of the focus.

References

[1] "Parabolas" by Math Open Reference
[2] "Quadratic Curves" by Wolfram MathWorld
[3] "Focus and Directrix" by Khan Academy
[4] "Axis of Symmetry" by Purplemath
[5] "Equation of a Parabola" by Mathway
[6] "Distance from the Vertex to the Focus" by GeoGebra

Introduction

In our previous article, we explored the properties of a parabola with a vertex at the origin and a directrix at y = 3. We found the equation of the parabola and used it to determine the value of a. We then used the formula for the distance from the vertex to the focus to find the coordinates of the focus. In this article, we will answer some frequently asked questions about parabolas and provide additional information to help you better understand the topic.

Q&A

Q: What is a parabola?

A: A parabola is a type of quadratic curve that has a specific shape. It is defined as the set of all points that are equidistant to the focus and the directrix.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola, depending on the orientation of the parabola.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is used to define the parabola and is located at a distance of 1/4a from the vertex along the axis of symmetry.

Q: How do I find the equation of a parabola?

A: To find the equation of a parabola, you need to know the value of a. You can use the formula for the distance from the vertex to the focus to find the value of a.

Q: How do I find the coordinates of the focus of a parabola?

A: To find the coordinates of the focus of a parabola, you need to know the value of a. You can use the formula for the distance from the vertex to the focus to find the coordinates of the focus.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix. It is used to define the parabola and is located at a distance of 1/4a from the vertex along the axis of symmetry.

Q: How do I determine the value of a for a parabola?

A: To determine the value of a for a parabola, you need to know the distance from the vertex to the directrix. You can use the formula for the distance from the vertex to the focus to find the value of a.

Q: What is the significance of the focus of a parabola?

A: The focus of a parabola is a fixed point that is used to define the parabola. It is located at a distance of 1/4a from the vertex along the axis of symmetry and is used to determine the shape of the parabola.

Additional Information

A parabola can be defined as the set of all points that are equidistant to the focus and the directrix.
The vertex of a parabola is the point where the parabola changes direction.
The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola.
The axis of symmetry of a parabola is the line that passes through the vertex and is perpendicular to the directrix.
The focus of a parabola is a fixed point that is used to define the parabola.
The value of a determines the shape of the parabola.