A Parabola Can Be Drawn Given A Focus Of $(7,10)$ And A Directrix Of $y=4$. What Can Be Said About The Parabola?

by ADMIN 117 views

=====================================================

Introduction

A parabola is a fundamental concept in mathematics, and it can be defined in various ways. One of the most common definitions is based on a focus and a directrix. In this article, we will explore what can be said about a parabola given a focus of $(7,10)$ and a directrix of $y=4$.

Understanding the Focus and Directrix

The focus of a parabola is a fixed point that is used to define the parabola. It is the point around which the parabola is symmetric. The directrix, on the other hand, is a line that is perpendicular to the axis of symmetry of the parabola. The directrix is used to define the shape of the parabola.

Properties of the Focus and Directrix

Distance between the Focus and the Directrix

The distance between the focus and the directrix is a critical property of the parabola. In this case, the focus is at $(7,10)$, and the directrix is at $y=4$. The distance between the focus and the directrix can be calculated as follows:

d=∣10−4∣=6d = |10 - 4| = 6

This means that the distance between the focus and the directrix is 6 units.

Axis of Symmetry

The axis of symmetry of the parabola is a line that passes through the focus and is perpendicular to the directrix. In this case, the axis of symmetry is a vertical line that passes through the point $(7,10)$.

Shape of the Parabola

The shape of the parabola is determined by the distance between the focus and the directrix. In this case, the distance is 6 units, which means that the parabola will be a relatively wide parabola.

Equation of the Parabola

The equation of a parabola in the form $(x-h)^2 = 4p(y-k)$, where $(h,k)$ is the vertex of the parabola, and $p$ is the distance between the focus and the directrix.

In this case, the vertex of the parabola is $(7,4)$, and the distance between the focus and the directrix is 6 units. Therefore, the equation of the parabola is:

(x−7)2=4(6)(y−4)(x-7)^2 = 4(6)(y-4)

Simplifying the equation, we get:

(x−7)2=24(y−4)(x-7)^2 = 24(y-4)

Conclusion

In conclusion, a parabola can be defined by a focus and a directrix. Given a focus of $(7,10)$ and a directrix of $y=4$, we can say that the distance between the focus and the directrix is 6 units, and the axis of symmetry is a vertical line that passes through the point $(7,10)$. The shape of the parabola is determined by the distance between the focus and the directrix, and the equation of the parabola can be written in the form $(x-h)^2 = 4p(y-k)$.

References

  • [1] "Parabola" by Math Open Reference. Retrieved 2023-12-01.
  • [2] "Focus and Directrix" by Wolfram MathWorld. Retrieved 2023-12-01.

Further Reading

  • [1] "Parabolas" by Khan Academy. Retrieved 2023-12-01.
  • [2] "Focus and Directrix" by MIT OpenCourseWare. Retrieved 2023-12-01.

Additional Resources

  • [1] "Parabola Calculator" by Calculator Soup. Retrieved 2023-12-01.
  • [2] "Focus and Directrix Calculator" by Mathway. Retrieved 2023-12-01.

=====================================================

Introduction

In our previous article, we explored the concept of a parabola defined by a focus and a directrix. We discussed the properties of the focus and directrix, the shape of the parabola, and the equation of the parabola. In this article, we will answer some frequently asked questions about parabolas defined by a focus and a directrix.

Q&A

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that is used to define the parabola. It is the point around which the parabola is symmetric.

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. It is used to define the shape of the parabola.

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

A: To find the distance between the focus and the directrix, you need to calculate the absolute value of the difference between the y-coordinates of the focus and the directrix.

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

A: The axis of symmetry of a parabola is a line that passes through the focus and is perpendicular to the directrix.

Q: How do I determine the shape of a parabola?

A: The shape of a parabola is determined by the distance between the focus and the directrix. A larger distance between the focus and the directrix results in a wider parabola.

Q: How do I write the equation of a parabola?

A: To write the equation of a parabola, you need to use the formula $(x-h)^2 = 4p(y-k)$, where $(h,k)$ is the vertex of the parabola, and $p$ is the distance between the focus and the directrix.

Q: Can a parabola have multiple foci?

A: No, a parabola can only have one focus.

Q: Can a parabola have multiple directrices?

A: No, a parabola can only have one directrix.

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

A: To find the vertex of a parabola, you need to find the midpoint between the focus and the directrix.

Q: How do I find the focal length of a parabola?

A: To find the focal length of a parabola, you need to divide the distance between the focus and the directrix by 2.

Conclusion

In conclusion, a parabola defined by a focus and a directrix is a fundamental concept in mathematics. By understanding the properties of the focus and directrix, the shape of the parabola, and the equation of the parabola, you can answer many questions about parabolas.

References

  • [1] "Parabola" by Math Open Reference. Retrieved 2023-12-01.
  • [2] "Focus and Directrix" by Wolfram MathWorld. Retrieved 2023-12-01.

Further Reading

  • [1] "Parabolas" by Khan Academy. Retrieved 2023-12-01.
  • [2] "Focus and Directrix" by MIT OpenCourseWare. Retrieved 2023-12-01.

Additional Resources

  • [1] "Parabola Calculator" by Calculator Soup. Retrieved 2023-12-01.
  • [2] "Focus and Directrix Calculator" by Mathway. Retrieved 2023-12-01.

Example Problems

  • Find the distance between the focus and the directrix of a parabola with a focus at $(3,5)$ and a directrix at $y=2$.
  • Write the equation of a parabola with a focus at $(4,6)$ and a directrix at $y=3$.
  • Find the vertex of a parabola with a focus at $(2,4)$ and a directrix at $y=1$.

Solutions

  • The distance between the focus and the directrix is $|5-2|=3$.
  • The equation of the parabola is $(x-4)^2=4(3)(y-3)$.
  • The vertex of the parabola is $(2,2.5)$.