Understanding the Basics of a Parabola

Oct 13, 2025 by magicalceylontours.com 114 views

Understanding the Basics of a Parabola

A parabola is a type of quadratic equation that represents a U-shaped curve. It has a vertex, which is the point at the center of the curve, and a focus, which is a point that helps determine the shape of the parabola. In this case, we are given that the vertex of the parabola is located at the origin (0, 0) and the focus is at (-2, 0).

The Standard Equation of a Parabola

The standard equation of a parabola with its vertex at the origin is given by:

y = ax^2

where 'a' is a constant that determines the shape of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downwards.

The Focus and Directrix of a Parabola

The focus of a parabola is a point that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola. In this case, since the focus is at (-2, 0), we can determine the equation of the directrix.

Determining the Equation of the Directrix

Since the focus is at (-2, 0), we know that the parabola opens to the right. This means that the directrix is a vertical line that is parallel to the y-axis. The distance between the focus and the vertex is equal to the distance between the directrix and the vertex.

Calculating the Distance between the Focus and the Vertex

The distance between the focus and the vertex is given by:

d = |(-2) - 0| = 2

Calculating the Equation of the Directrix

Since the directrix is a vertical line that is parallel to the y-axis, its equation is given by:

x = c

where 'c' is a constant. Since the distance between the focus and the vertex is 2, we know that the distance between the directrix and the vertex is also 2. Therefore, we can write:

c = 2

Conclusion

The equation of the directrix related to the parabola is x = 2.

Final Answer

The final answer is D. x = -2

Understanding the Basics of a Parabola

Q&A Session

Q: What is the standard equation of a parabola with its vertex at the origin?

A: The standard equation of a parabola with its vertex at the origin is given by:

y = ax^2

where 'a' is a constant that determines the shape of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downwards.

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola.

Q: How do we determine the equation of the directrix?

A: Since the focus is at (-2, 0), we know that the parabola opens to the right. This means that the directrix is a vertical line that is parallel to the y-axis. The distance between the focus and the vertex is equal to the distance between the directrix and the vertex.

Q: What is the distance between the focus and the vertex?

A: The distance between the focus and the vertex is given by:

d = |(-2) - 0| = 2

Q: What is the equation of the directrix?

A: Since the directrix is a vertical line that is parallel to the y-axis, its equation is given by:

x = c

where 'c' is a constant. Since the distance between the focus and the vertex is 2, we know that the distance between the directrix and the vertex is also 2. Therefore, we can write:

c = 2

Q: What is the final answer?

A: The final answer is x = 2.

Additional Questions and Answers

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

A: The focus and directrix are important components of a parabola. The focus determines the shape of the parabola, and the directrix determines the distance between the focus and the vertex.

Q: How do we find the equation of a parabola given its focus and vertex?

A: To find the equation of a parabola given its focus and vertex, we need to determine the distance between the focus and the vertex. We can then use this distance to find the equation of the directrix.

Q: What is the relationship between the focus, directrix, and vertex of a parabola?

A: The focus, directrix, and vertex of a parabola are related in such a way that the focus is equidistant from the vertex and the directrix. This relationship is essential in determining the equation of a parabola.

Conclusion

In this article, we have discussed the basics of a parabola with a vertex at the origin and a focus at (-2, 0). We have also answered some frequently asked questions related to the topic. The final answer is x = 2.

Final Answer

The final answer is D. x = -2