A Parabola Can Be Drawn Given A Focus Of $(-4,-9)$ And A Directrix Of $y=-3$. Write The Equation Of The Parabola In Any Form.

Introduction

A parabola is a fundamental concept in mathematics, and it can be defined in various ways. One of the most interesting and unique ways to define a parabola is by specifying its focus and directrix. In this article, we will explore how to write the equation of a parabola given its focus and directrix. We will use the focus $(-4,-9)$ and the directrix $y=-3$ as an example.

Understanding the Focus and Directrix

The focus of a parabola is a fixed point that is equidistant from the parabola and its directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola. In this case, the focus is located at $(-4,-9)$, and the directrix is the line $y=-3$. The axis of symmetry of the parabola is the vertical line that passes through the focus.

The Standard Equation of a Parabola

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

$$y^2 = 4ax$$

where $a$ is the distance from the vertex to the focus. However, in this case, the vertex is not at the origin, and the parabola is not in the standard position. We need to use a different approach to find the equation of the parabola.

Using the Focus and Directrix to Find the Equation

To find the equation of the parabola, we can use the fact that the focus is equidistant from the parabola and its directrix. Let's call the distance from the focus to the directrix $p$. Then, the distance from the focus to the parabola is also $p$. We can use this information to find the equation of the parabola.

Finding the Distance from the Focus to the Directrix

The distance from the focus $(-4,-9)$ to the directrix $y=-3$ is given by:

$$p = \left| -9 - (-3) \right| = 6$$

Finding the Equation of the Parabola

Now that we have the distance from the focus to the directrix, we can use it to find the equation of the parabola. The equation of a parabola with its focus at $(h,k)$ and its directrix $y=k-p$ is given by:

$$\left( y - k \right)^2 = 4p \left( x - h \right)$$

In this case, the focus is at $(-4,-9)$, and the directrix is $y=-3$. Plugging in the values, we get:

$$\left( y - (-9) \right)^2 = 4(6) \left( x - (-4) \right)$$

Simplifying the equation, we get:

$$\left( y + 9 \right)^2 = 24 \left( x + 4 \right)$$

Converting to Vertex Form

We can convert the equation to vertex form by completing the square:

$$\left( y + 9 \right)^2 = 24 \left( x + 4 \right)$$

$$y^2 + 18y + 81 = 24x + 96$$

$$y^2 + 18y - 24x + 81 - 96 = 0$$

$$y^2 + 18y - 24x - 15 = 0$$

Converting to Standard Form

We can convert the equation to standard form by expanding the squared term:

$$y^2 + 18y - 24x - 15 = 0$$

$$y^2 + 18y = 24x + 15$$

$$y^2 + 18y - 24x - 15 = 0$$

Conclusion

In this article, we have shown how to write the equation of a parabola given its focus and directrix. We used the focus $(-4,-9)$ and the directrix $y=-3$ as an example and derived the equation of the parabola in various forms. We hope that this article has provided a clear and concise explanation of how to find the equation of a parabola given its focus and directrix.

References

• [1] "Parabola" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Focus and Directrix of a Parabola" by Purplemath. Retrieved 2023-02-20.
• [3] "Equation of a Parabola" by Math Is Fun. Retrieved 2023-02-20.

Further Reading

• [1] "Parabolas" by Khan Academy. Retrieved 2023-02-20.
• [2] "Focus and Directrix" by Wolfram MathWorld. Retrieved 2023-02-20.
• [3] "Equation of a Parabola" by MIT OpenCourseWare. Retrieved 2023-02-20.
  

Introduction

In our previous article, we explored how to write the equation of a parabola given its focus and directrix. We used the focus $(-4,-9)$ and the directrix $y=-3$ as an example and derived the equation of the parabola in various forms. In this article, we will answer some of the most frequently asked questions about parabolas defined by a focus and a directrix.

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is equidistant from the parabola and its directrix.

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 I find the equation of a parabola given its focus and directrix?

A: To find the equation of a parabola given its focus and directrix, you can use the formula:

$$\left( y - k \right)^2 = 4p \left( x - h \right)$$

where $(h,k)$ is the focus and $p$ is the distance from the focus to the directrix.

Q: What is the distance from the focus to the directrix?

A: The distance from the focus to the directrix is given by:

$$p = \left| k - (k-p) \right|$$

Q: How do I convert the equation of a parabola from vertex form to standard form?

A: To convert the equation of a parabola from vertex form to standard form, you can expand the squared term and simplify the equation.

Q: How do I convert the equation of a parabola from standard form to vertex form?

A: To convert the equation of a parabola from standard form to vertex form, you can complete the square and simplify the equation.

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

A: The axis of symmetry of a parabola is the vertical line that passes through the focus.

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

A: To find the vertex of a parabola, you can use the formula:

$$\left( h, k \right)$$

where $(h,k)$ is the focus.

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

A: The focus and the directrix of a parabola are equidistant from the parabola.

Q: How do I graph a parabola given its focus and directrix?

A: To graph a parabola given its focus and directrix, you can use the following steps:

1. Plot the focus and the directrix on a coordinate plane.
2. Draw a line through the focus that is perpendicular to the directrix.
3. Draw a line through the directrix that is perpendicular to the axis of symmetry.
4. The parabola is the set of all points that are equidistant from the focus and the directrix.

Conclusion

In this article, we have answered some of the most frequently asked questions about parabolas defined by a focus and a directrix. We hope that this article has provided a clear and concise explanation of the concepts and formulas involved.

References

• [1] "Parabola" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Focus and Directrix of a Parabola" by Purplemath. Retrieved 2023-02-20.
• [3] "Equation of a Parabola" by Math Is Fun. Retrieved 2023-02-20.

Further Reading

• [1] "Parabolas" by Khan Academy. Retrieved 2023-02-20.
• [2] "Focus and Directrix" by Wolfram MathWorld. Retrieved 2023-02-20.
• [3] "Equation of a Parabola" by MIT OpenCourseWare. Retrieved 2023-02-20.