Find The Equation For The Parabola That Has Its Focus At \left(-\frac{53}{4}, -9\right ] And Has A Directrix At X = 37 4 X = \frac{37}{4} X = 4 37 ​ . Equation: □ \square □

Introduction

In mathematics, a parabola is a type of quadratic curve that has a specific focus and directrix. The focus is a fixed point that is equidistant from the directrix and the curve, while the directrix is a fixed line that the curve approaches but never touches. In this article, we will explore how to find the equation of a parabola given its focus and directrix.

Understanding the Focus and Directrix

The focus of a parabola is a point that is equidistant from the directrix and the curve. This means that if we draw a line from the focus to the curve, it will be perpendicular to the directrix. The directrix is a fixed line that the curve approaches but never touches. In this case, the focus is located at $\left(-\frac{53}{4}, -9\right)$, and the directrix is located at $x = \frac{37}{4}$.

The Standard Equation of a Parabola

The standard equation of a parabola is given by:

$$y = \frac{1}{4p}(x - h)^2 + k$$

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

Finding the Equation of the Parabola

To find the equation of the parabola, we need to determine the values of $h$, $k$, and $p$. Since the focus is located at $\left(-\frac{53}{4}, -9\right)$, we know that the vertex is located at the midpoint of the focus and the directrix. The midpoint of the focus and the directrix is given by:

$$\left(\frac{-\frac{53}{4} + \frac{37}{4}}{2}, -9\right) = \left(-\frac{8}{4}, -9\right) = (-2, -9)$$

Therefore, the vertex of the parabola is located at $(-2, -9)$.

Determining the Value of $p$

The value of $p$ is the distance from the vertex to the focus. Since the vertex is located at $(-2, -9)$ and the focus is located at $\left(-\frac{53}{4}, -9\right)$, we can calculate the distance between the two points using the distance formula:

$$p = \sqrt{\left(-\frac{53}{4} + 2\right)^2 + (0)^2} = \sqrt{\left(-\frac{45}{4}\right)^2} = \frac{45}{4}$$

Writing the Equation of the Parabola

Now that we have determined the values of $h$, $k$, and $p$, we can write the equation of the parabola using the standard equation:

$$y = \frac{1}{4p}(x - h)^2 + k$$

Substituting the values of $h$, $k$, and $p$, we get:

$$y = \frac{1}{4\left(\frac{45}{4}\right)}(x + 2)^2 - 9$$

Simplifying the equation, we get:

$$y = \frac{1}{45}(x + 2)^2 - 9$$

Conclusion

In this article, we have explored how to find the equation of a parabola given its focus and directrix. We have determined the values of $h$, $k$, and $p$ using the given information, and have written the equation of the parabola using the standard equation. The equation of the parabola is given by:

$$y = \frac{1}{45}(x + 2)^2 - 9$$

This equation represents the parabola with a focus at $\left(-\frac{53}{4}, -9\right)$ and a directrix at $x = \frac{37}{4}$.

References

• [1] "Parabolas" by Math Open Reference. Retrieved from https://www.mathopenref.com/parabola.html
• [2] "Equation of a Parabola" by Purplemath. Retrieved from https://www.purplemath.com/modules/parabola.htm

Discussion

• What is the vertex of the parabola?
• How do you determine the value of $p$?
• What is the standard equation of a parabola?
• How do you write the equation of a parabola given its focus and directrix?

Related Topics

• Equation of a Parabola
• Vertex of a Parabola
• Focus and Directrix of a Parabola

Tags

• Parabola
• Focus
• Directrix
• Vertex
• Equation
• Mathematics