Properties of the Directrix

Oct 13, 2025 by magicalceylontours.com 51 views

Introduction

A parabola is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will explore the properties of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis. We will analyze the possible equations of such a parabola and determine which one is correct.

Understanding Parabolas

A parabola is a quadratic curve that 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 curve changes direction, and it is the lowest or highest point on the curve. In this case, the vertex is located at (0,0), which means that the parabola opens either upwards or downwards.

Properties of the Directrix

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. In this case, the directrix crosses the negative part of the y-axis, which means that it is a horizontal line with a negative y-coordinate. This implies that the parabola opens downwards.

Possible Equations of the Parabola

There are four possible equations of the parabola, and we need to determine which one is correct. Let's analyze each option:

Option A: x^2 = -4y

This equation represents a parabola that opens downwards. The vertex of the parabola is located at (0,0), and the directrix is a horizontal line with a negative y-coordinate. This option seems to satisfy the given conditions.

Option B: x^2 = 4y

This equation represents a parabola that opens upwards. The vertex of the parabola is located at (0,0), but the directrix is a horizontal line with a positive y-coordinate. This option does not satisfy the given conditions.

Option C: y^2 = 4x

This equation represents a parabola that opens to the right. The vertex of the parabola is located at (0,0), but the directrix is a vertical line with a negative x-coordinate. This option does not satisfy the given conditions.

Option D: y^2 = -4x

This equation represents a parabola that opens to the left. The vertex of the parabola is located at (0,0), but the directrix is a vertical line with a positive x-coordinate. This option does not satisfy the given conditions.

Conclusion

Based on the analysis of the possible equations of the parabola, we can conclude that the correct equation is:

x^2 = -4y

This equation represents a parabola that opens downwards, with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis.

Final Answer

The final answer is A. x^2 = -4y.

Introduction

In our previous article, we explored the properties of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis. We analyzed the possible equations of such a parabola and determined that the correct equation is x^2 = -4y. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the definition of a parabola?

A: A parabola is a quadratic curve that can be 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 curve changes direction, and it is the lowest or highest point on the curve.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola.

Q: How do you determine the equation of a parabola?

A: To determine the equation of a parabola, you need to know the coordinates of the vertex and the direction of the axis of symmetry.

Q: What is the significance of the directrix in a parabola?

A: The directrix is a line that is perpendicular to the axis of symmetry of the parabola, and it plays a crucial role in defining the shape of the parabola.

Q: Can a parabola have a directrix that crosses the positive part of the y-axis?

A: No, a parabola with a vertex at (0,0) can only have a directrix that crosses the negative part of the y-axis.

Q: What is the equation of a parabola that opens upwards?

A: The equation of a parabola that opens upwards is x^2 = 4y.

Q: What is the equation of a parabola that opens to the right?

A: The equation of a parabola that opens to the right is y^2 = 4x.

Q: What is the equation of a parabola that opens to the left?

A: The equation of a parabola that opens to the left is y^2 = -4x.

Q: Can a parabola have a vertex at (0,0) and a directrix that crosses the positive part of the x-axis?

A: No, a parabola with a vertex at (0,0) can only have a directrix that crosses the negative part of the y-axis or the positive part of the x-axis, but not both.

Conclusion

In this article, we answered some frequently asked questions related to the topic of a parabola with a vertex at (0,0) and a directrix that crosses the negative part of the y-axis. We hope that this article has provided you with a better understanding of the properties of parabolas and their equations.

Final Answer

The final answer is A. x^2 = -4y.