Finding the Distance Between the Focus and the Vertex

Oct 13, 2025 by ludoplayers7.com 138 views

Introduction

In mathematics, a parabola is a fundamental concept that can be described using various forms, including standard, vertex, and focus-directrix forms. The focus-directrix form is particularly useful when the focus and directrix of a parabola are known, as it allows us to derive the equation of the parabola and find its vertex coordinates. In this article, we will explore how to find the vertex coordinates of a parabola given its focus and directrix.

Understanding the Focus-Directrix Form

The focus-directrix form of a parabola is given by the equation:

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

where $(h, k)$ is the vertex of the parabola, and $p$ is the distance between the focus and the vertex. The focus is located at $(h + p, k)$ , and the directrix is a horizontal line at a distance $p$ from the vertex.

Given Information

We are given that the focus of the parabola is located at $(-2, -4)$ and the directrix is a horizontal line at $y = -5$ . We need to find the coordinates of the vertex.

Finding the Distance Between the Focus and the Vertex

To find the distance between the focus and the vertex, we can use the distance formula:

p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where $(x_1, y_1)$ is the vertex, and $(x_2, y_2)$ is the focus. In this case, we have:

p = \sqrt{(-2 - h)^2 + (-4 - k)^2}

However, we are not given the values of $h$ and $k$ . We need to find them first.

Finding the Vertex Coordinates

Since the directrix is a horizontal line at $y = -5$ , we know that the vertex must be at a distance $p$ from the directrix. This means that the $y$ -coordinate of the vertex is $k = -5 + p$ .

We can substitute this expression for $k$ into the equation for $p$ :

p = \sqrt{(-2 - h)^2 + (-5 + p)^2}

Simplifying this equation, we get:

p = \sqrt{(-2 - h)^2 + (-5 + p)^2}

p^2 = (-2 - h)^2 + (-5 + p)^2

Expanding and simplifying, we get:

p^2 = h^2 + 4h + 4 + 25 - 10p + p^2

Subtracting $p^2$ from both sides, we get:

0 = h^2 + 4h + 29 - 10p

Rearranging this equation, we get:

h^2 + 4h + (29 - 10p) = 0

This is a quadratic equation in $h$ . We can solve it using the quadratic formula:

h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, we have:

a = 1, b = 4, c = 29 - 10p

Substituting these values into the quadratic formula, we get:

h = \frac{-4 \pm \sqrt{16 - 4(29 - 10p)}}{2}

Simplifying this expression, we get:

h = \frac{-4 \pm \sqrt{16 - 116 + 40p}}{2}

h = \frac{-4 \pm \sqrt{-100 + 40p}}{2}

Since the square root of a negative number is not a real number, we must have:

-100 + 40p \geq 0

Solving this inequality, we get:

p \geq \frac{100}{40}

p \geq 2.5

Since $p$ is the distance between the focus and the vertex, it must be a positive number. Therefore, we can conclude that:

p = 2.5

Now that we have found the value of $p$ , we can substitute it into the equation for $h$ :

h = \frac{-4 \pm \sqrt{-100 + 40(2.5)}}{2}

Simplifying this expression, we get:

h = \frac{-4 \pm \sqrt{-100 + 100}}{2}

h = \frac{-4 \pm \sqrt{0}}{2}

h = \frac{-4}{2}

h = -2

Now that we have found the value of $h$ , we can find the value of $k$ :

k = -5 + p

k = -5 + 2.5

k = -2.5

Therefore, the coordinates of the vertex are:

(h, k) = (-2, -2.5)

Conclusion

In this article, we have shown how to find the vertex coordinates of a parabola given its focus and directrix. We used the focus-directrix form of the parabola and the distance formula to find the distance between the focus and the vertex. We then used the quadratic formula to solve for the $x$ -coordinate of the vertex. Finally, we found the $y$ -coordinate of the vertex using the equation for the directrix. The coordinates of the vertex are $(-2, -2.5)$ .

Introduction

In our previous article, we explored how to find the vertex coordinates of a parabola given its focus and directrix. We used the focus-directrix form of the parabola and the distance formula to find the distance between the focus and the vertex. We then used the quadratic formula to solve for the $x$ -coordinate of the vertex. Finally, we found the $y$ -coordinate of the vertex using the equation for the directrix. In this article, we will answer some frequently asked questions about finding the vertex coordinates of a parabola given its focus and directrix.

Q: What is the focus-directrix form of a parabola?

A: The focus-directrix form of a parabola is given by the equation:

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

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

Q: How do I find the distance between the focus and the vertex?

A: To find the distance between the focus and the vertex, you can use the distance formula:

p = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

where $(x_1, y_1)$ is the vertex, and $(x_2, y_2)$ is the focus.

Q: What if the directrix is not a horizontal line?

A: If the directrix is not a horizontal line, you will need to use a different approach to find the vertex coordinates. In this case, you can use the equation of the directrix to find the $y$ -coordinate of the vertex.

Q: Can I use the focus-directrix form to find the equation of the parabola?

A: Yes, you can use the focus-directrix form to find the equation of the parabola. Once you have found the vertex coordinates, you can substitute them into the equation to find the equation of the parabola.

Q: What if I have a parabola with a vertical axis of symmetry?

A: If you have a parabola with a vertical axis of symmetry, you will need to use a different approach to find the vertex coordinates. In this case, you can use the equation of the directrix to find the $x$ -coordinate of the vertex.

Q: Can I use the focus-directrix form to find the vertex coordinates of a parabola with a vertical axis of symmetry?

A: Yes, you can use the focus-directrix form to find the vertex coordinates of a parabola with a vertical axis of symmetry. However, you will need to use a different approach to find the $x$ -coordinate of the vertex.

Q: What if I have a parabola with a horizontal axis of symmetry and a directrix that is not a horizontal line?

A: If you have a parabola with a horizontal axis of symmetry and a directrix that is not a horizontal line, you will need to use a different approach to find the vertex coordinates. In this case, you can use the equation of the directrix to find the $y$ -coordinate of the vertex.

Q: Can I use the focus-directrix form to find the equation of a parabola with a vertical axis of symmetry?

A: Yes, you can use the focus-directrix form to find the equation of a parabola with a vertical axis of symmetry. However, you will need to use a different approach to find the equation of the parabola.

Q: What if I have a parabola with a vertical axis of symmetry and a directrix that is not a horizontal line?

A: If you have a parabola with a vertical axis of symmetry and a directrix that is not a horizontal line, you will need to use a different approach to find the vertex coordinates. In this case, you can use the equation of the directrix to find the $x$ -coordinate of the vertex.

Conclusion

In this article, we have answered some frequently asked questions about finding the vertex coordinates of a parabola given its focus and directrix. We have shown how to use the focus-directrix form to find the vertex coordinates of a parabola with a horizontal axis of symmetry and a directrix that is a horizontal line. We have also shown how to use the equation of the directrix to find the vertex coordinates of a parabola with a vertical axis of symmetry and a directrix that is not a horizontal line.