Understanding the Given Parabola

Oct 13, 2025 by pynkocean.com 149 views

Introduction

A parabola is a type of quadratic equation that can be represented in various forms, including the standard form, vertex form, and focus-directrix form. In this article, we will focus on finding the focus and directrix of a parabola given in the vertex form. The vertex form of a parabola is represented as ${(x-h)^2 = 4p(y-k)}$ , where ${(h,k)}$ is the vertex of the parabola and ${p}$ is the distance from the vertex to the focus and directrix.

Understanding the Given Parabola

The given parabola is ${(x-5)^2 = 8(y-6)}$ . To find the focus and directrix, we need to rewrite the equation in the standard form of a parabola, which is ${(x-h)^2 = 4p(y-k)}$ . By comparing the given equation with the standard form, we can see that the vertex of the parabola is ${(5,6)}$ and the coefficient of ${(y-6)}$ is ${8}$ .

Rewriting the Equation in Standard Form

To rewrite the equation in standard form, we need to isolate the term with the variable ${y}$ . We can do this by dividing both sides of the equation by ${8}$ .

${(x-5)^2 = 8(y-6)}$ ${\frac{(x-5)^2}{8} = y-6}$ ${y = \frac{(x-5)^2}{8} + 6}$

Now that we have the equation in standard form, we can identify the values of ${h}$ , ${k}$ , and ${p}$ . The value of ${h}$ is ${5}$ , the value of ${k}$ is ${6}$ , and the value of ${p}$ is ${\sqrt{\frac{8}{4}} = \sqrt{2}}$ .

Finding the Focus

The focus of a parabola is located at a distance of ${p}$ units from the vertex, either above or below it, depending on the sign of ${p}$ . Since the value of ${p}$ is positive, the focus is located above the vertex.

The coordinates of the focus are ${(h, k+p)}$ . Substituting the values of ${h}$ , ${k}$ , and ${p}$ , we get:

Focus: ${(5, 6+\sqrt{2})}$

Finding the Directrix

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of ${p}$ units from the vertex. Since the value of ${p}$ is positive, the directrix is located below the vertex.

The equation of the directrix is ${y = k-p}$ . Substituting the values of ${k}$ and ${p}$ , we get:

Directrix: ${y = 6-\sqrt{2}}$

Conclusion

In this article, we have found the focus and directrix of a parabola given in the vertex form. We have rewritten the equation in standard form, identified the values of ${h}$ , ${k}$ , and ${p}$ , and used these values to find the coordinates of the focus and the equation of the directrix. The focus is located at ${(5, 6+\sqrt{2})}$ and the directrix is ${y = 6-\sqrt{2}}$ .

Key Takeaways

The vertex form of a parabola is represented as ${(x-h)^2 = 4p(y-k)}$ .
The focus of a parabola is located at a distance of ${p}$ units from the vertex, either above or below it.
The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of ${p}$ units from the vertex.
The equation of the directrix is ${y = k-p}$ .

Practice Problems

Find the focus and directrix of the parabola ${(x-2)^2 = 4(y-3)}$ .
Find the focus and directrix of the parabola ${(x+1)^2 = 8(y+2)}$ .

Solutions

Focus: ${(2, 3+\sqrt{2})}$ Directrix: ${y = 3-\sqrt{2}}$
Focus: ${(-1, -2+\sqrt{2})}$ Directrix: ${y = -2-\sqrt{2}}$
Frequently Asked Questions (FAQs) about Finding the Focus and Directrix of a Parabola =====================================================================================

Q: What is the vertex form of a parabola?

A: The vertex form of a parabola is represented as ${(x-h)^2 = 4p(y-k)}$ , where ${(h,k)}$ is the vertex of the parabola and ${p}$ is the distance from the vertex to the focus and directrix.

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

A: To find the focus of a parabola, you need to identify the values of ${h}$ , ${k}$ , and ${p}$ from the vertex form of the parabola. The focus is located at a distance of ${p}$ units from the vertex, either above or below it, depending on the sign of ${p}$ . The coordinates of the focus are ${(h, k+p)}$ .

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

A: To find the directrix of a parabola, you need to identify the values of ${h}$ , ${k}$ , and ${p}$ from the vertex form of the parabola. The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of ${p}$ units from the vertex. The equation of the directrix is ${y = k-p}$ .

Q: What is the significance of the value of ${p}$ in the vertex form of a parabola?

A: The value of ${p}$ in the vertex form of a parabola represents the distance from the vertex to the focus and directrix. It is a critical component in determining the location of the focus and directrix.

Q: Can the focus and directrix of a parabola be located at the same point?

A: No, the focus and directrix of a parabola cannot be located at the same point. The focus is located at a distance of ${p}$ units from the vertex, either above or below it, while the directrix is located at a distance of ${p}$ units from the vertex, but on the opposite side.

Q: How do I determine the orientation of the parabola (upward or downward)?

A: To determine the orientation of the parabola, you need to examine the sign of the value of ${p}$ . If ${p}$ is positive, the parabola opens upward. If ${p}$ is negative, the parabola opens downward.

Q: Can the parabola be a degenerate case (i.e., a single point or a line)?

A: Yes, the parabola can be a degenerate case, which means it can be a single point or a line. This occurs when the value of ${p}$ is zero.

Q: How do I find the equation of the directrix in the case of a degenerate parabola?

A: In the case of a degenerate parabola, the equation of the directrix is not defined. This is because the directrix is a line that is perpendicular to the axis of symmetry of the parabola, and in the case of a degenerate parabola, the axis of symmetry is not defined.

Q: Can the focus and directrix of a parabola be located at infinity?

A: No, the focus and directrix of a parabola cannot be located at infinity. The focus and directrix are always located at a finite distance from the vertex of the parabola.

Q: How do I find the equation of the directrix in the case of a parabola with a vertical axis of symmetry?

A: In the case of a parabola with a vertical axis of symmetry, the equation of the directrix is ${x = h-p}$ .