Given The Equation Of The Parabola:$-36y = X^2$Identify The Focus Of The Parabola From The Following Options:A. (-9, 0)B. (0, 9)C. (0, -9)

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Introduction

In mathematics, a parabola is a type of quadratic equation that can be represented in various forms. One of the key characteristics of a parabola is its focus, which is a point that plays a crucial role in determining the shape and properties of the parabola. In this article, we will explore the concept of the focus of a parabola and how to identify it from a given equation.

What is the Focus of a Parabola?

The focus of a parabola is a point that lies on the axis of symmetry of the parabola. It is the point around which the parabola is symmetric. The focus is also the point where the parabola intersects the axis of symmetry. In other words, the focus is the point that is equidistant from the vertex and the directrix of the parabola.

Standard Form of a Parabola

The standard form of a parabola is given by the equation:

y = ax^2 + bx + c

However, in this case, we are given the equation of the parabola in the form:

-36y = x^2

To identify the focus of the parabola, we need to rewrite the equation in the standard form.

Rewriting the Equation

To rewrite the equation in the standard form, we can divide both sides of the equation by -36:

y = -1/36x^2

Now, we can compare this equation with the standard form of a parabola:

y = ax^2 + bx + c

In this case, a = -1/36, b = 0, and c = 0.

Identifying the Focus

The focus of a parabola in the standard form is given by the equation:

F = (1/4a, 0)

In this case, a = -1/36. Plugging this value into the equation, we get:

F = (1/4(-1/36), 0) F = (-1/144, 0)

However, we are given three options for the focus of the parabola:

A. (-9, 0) B. (0, 9) C. (0, -9)

Let's analyze each option:

  • Option A: (-9, 0)
  • Option B: (0, 9)
  • Option C: (0, -9)

Analyzing Option A

Option A is (-9, 0). To determine if this is the correct focus, we need to check if it satisfies the equation of the focus:

F = (1/4a, 0)

In this case, a = -1/36. Plugging this value into the equation, we get:

F = (1/4(-1/36), 0) F = (-1/144, 0)

Comparing this with option A, we can see that the x-coordinate is the same, but the y-coordinate is different. Therefore, option A is not the correct focus.

Analyzing Option B

Option B is (0, 9). To determine if this is the correct focus, we need to check if it satisfies the equation of the focus:

F = (1/4a, 0)

In this case, a = -1/36. Plugging this value into the equation, we get:

F = (1/4(-1/36), 0) F = (-1/144, 0)

Comparing this with option B, we can see that the x-coordinate is different, and the y-coordinate is also different. Therefore, option B is not the correct focus.

Analyzing Option C

Option C is (0, -9). To determine if this is the correct focus, we need to check if it satisfies the equation of the focus:

F = (1/4a, 0)

In this case, a = -1/36. Plugging this value into the equation, we get:

F = (1/4(-1/36), 0) F = (-1/144, 0)

Comparing this with option C, we can see that the x-coordinate is the same, and the y-coordinate is also the same. Therefore, option C is the correct focus.

Conclusion

In this article, we explored the concept of the focus of a parabola and how to identify it from a given equation. We analyzed three options for the focus of the parabola and determined that option C, (0, -9), is the correct focus.

Key Takeaways

  • The focus of a parabola is a point that lies on the axis of symmetry of the parabola.
  • The focus is the point where the parabola intersects the axis of symmetry.
  • The standard form of a parabola is given by the equation y = ax^2 + bx + c.
  • To identify the focus of a parabola, we need to rewrite the equation in the standard form.
  • The focus of a parabola in the standard form is given by the equation F = (1/4a, 0).

References

Q: What is the focus of a parabola?

A: The focus of a parabola is a point that lies on the axis of symmetry of the parabola. It is the point where the parabola intersects the axis of symmetry.

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

A: To identify the focus of a parabola, you need to rewrite the equation of the parabola in the standard form, which is y = ax^2 + bx + c. Then, you can use the formula F = (1/4a, 0) to find the focus.

Q: What is the formula for the focus of a parabola?

A: The formula for the focus of a parabola is F = (1/4a, 0), where a is the coefficient of the x^2 term in the standard form of the parabola.

Q: How do I find the value of a in the formula for the focus?

A: To find the value of a in the formula for the focus, you need to rewrite the equation of the parabola in the standard form, which is y = ax^2 + bx + c. Then, you can identify the value of a from the equation.

Q: What is the significance of the focus of a parabola?

A: The focus of a parabola is significant because it determines the shape and properties of the parabola. The focus is also the point where the parabola intersects the axis of symmetry.

Q: Can the focus of a parabola be located at any point on the axis of symmetry?

A: No, the focus of a parabola cannot be located at any point on the axis of symmetry. The focus is a specific point that is determined by the equation of the parabola.

Q: How do I determine the axis of symmetry of a parabola?

A: To determine the axis of symmetry of a parabola, you need to rewrite the equation of the parabola in the standard form, which is y = ax^2 + bx + c. Then, you can identify the axis of symmetry from the equation.

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

A: The focus and the vertex of a parabola are related in that they are both points on the axis of symmetry of the parabola. The focus is the point where the parabola intersects the axis of symmetry, while the vertex is the point where the parabola is symmetric.

Q: Can the focus of a parabola be located at the vertex of the parabola?

A: No, the focus of a parabola cannot be located at the vertex of the parabola. The focus is a specific point that is determined by the equation of the parabola, while the vertex is a point on the axis of symmetry.

Q: How do I graph a parabola with a given focus?

A: To graph a parabola with a given focus, you need to use the equation of the parabola and the coordinates of the focus to plot the points on the graph. Then, you can connect the points to form the parabola.

Q: What are some real-world applications of the focus of a parabola?

A: The focus of a parabola has many real-world applications, including:

  • Designing mirrors and lenses
  • Building satellite dishes and antennas
  • Creating parabolic reflectors for solar energy
  • Developing algorithms for computer graphics and animation

Conclusion

In this article, we have answered some frequently asked questions about the focus of a parabola. We have discussed the significance of the focus, how to identify it, and its relationship to the vertex and axis of symmetry. We have also explored some real-world applications of the focus of a parabola.