Find The Focus And Directrix Of The Following Parabola:$ (y-6)^2=12(x-2)}$Focus ([?], [?])Directrix: { X =$ $ [?]

Mar 13, 2025 by ADMIN 115 views

Find the Focus and Directrix of a Parabola

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Introduction

A parabola is a fundamental concept in mathematics, particularly in the field of algebra and geometry. It is a U-shaped curve that can be defined by a quadratic equation in two variables. In this article, we will focus on finding the focus and directrix of a given parabola, which is a crucial aspect of understanding the properties of parabolas.

What is a Parabola?

A parabola is a quadratic curve that can be defined by the equation:

y = ax^2 + bx + c

x = ay^2 + by + c

where $a$ , $b$ , and $c$ are constants. The parabola can be opened upwards or downwards, depending on the sign of $a$ . If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

Standard Form of a Parabola

The standard form of a parabola is given by:

y - k = a(x - h)^2

x - k = a(y - h)^2

where $(h, k)$ is the vertex of the parabola. The vertex is the point on the parabola that is equidistant from the focus and the directrix.

Focus and Directrix of a Parabola

The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $1/(4a)$ from the vertex.

Finding the Focus and Directrix of a Parabola

To find the focus and directrix of a parabola, we need to rewrite the equation of the parabola in standard form. Let's consider the given parabola:

(y-6)^2=12(x-2)

We can rewrite this equation as:

y^2 - 12x + 12y + 36 = 0

Completing the square, we get:

(y - 6)^2 = 12(x - 2)

This is the standard form of a parabola, where the vertex is $(2, 6)$ and the equation is:

y - 6 = 12(x - 2)

Calculating the Focus and Directrix

The focus of the parabola is located at a distance of $1/(4a)$ from the vertex. In this case, $a = 12$ , so the focus is located at:

\left(2 + \frac{1}{48}, 6\right)

The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $1/(4a)$ from the vertex. In this case, the directrix is a vertical line located at:

x = 2 - \frac{1}{48}

Conclusion

In this article, we have discussed the concept of a parabola and its focus and directrix. We have also provided a step-by-step guide on how to find the focus and directrix of a given parabola. The focus and directrix of a parabola are crucial aspects of understanding the properties of parabolas, and this article has provided a comprehensive overview of these concepts.

Example Problems

Problem 1

Find the focus and directrix of the parabola:

(x-3)^2=8(y-2)

Solution

We can rewrite this equation as:

x^2 - 8y + 8x + 12 = 0

Completing the square, we get:

(x - 3)^2 = 8(y - 2)

This is the standard form of a parabola, where the vertex is $(3, 2)$ and the equation is:

x - 3 = 8(y - 2)

The focus of the parabola is located at a distance of $1/(4a)$ from the vertex. In this case, $a = 8$ , so the focus is located at:

\left(3 + \frac{1}{32}, 2\right)

The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $1/(4a)$ from the vertex. In this case, the directrix is a horizontal line located at:

y = 2 - \frac{1}{32}

Problem 2

Find the focus and directrix of the parabola:

(y-4)^2=16(x-1)

Solution

We can rewrite this equation as:

y^2 - 16x + 16y + 16 = 0

Completing the square, we get:

(y - 4)^2 = 16(x - 1)

This is the standard form of a parabola, where the vertex is $(1, 4)$ and the equation is:

y - 4 = 16(x - 1)

The focus of the parabola is located at a distance of $1/(4a)$ from the vertex. In this case, $a = 16$ , so the focus is located at:

\left(1 + \frac{1}{64}, 4\right)

x = 1 - \frac{1}{64}

Final Thoughts

In conclusion, finding the focus and directrix of a parabola is a crucial aspect of understanding the properties of parabolas. By following the steps outlined in this article, you can find the focus and directrix of any given parabola. Remember to rewrite the equation of the parabola in standard form, and then use the formula for the focus and directrix to find the desired values. With practice and patience, you will become proficient in finding the focus and directrix of parabolas.

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Introduction

In our previous article, we discussed the concept of parabolas and how to find the focus and directrix of a given parabola. However, we understand that there may be many questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about parabolas.

Q&A

Q1: What is a parabola?

A1: A parabola is a quadratic curve that can be defined by a quadratic equation in two variables. It is a U-shaped curve that can be opened upwards or downwards, depending on the sign of the coefficient of the squared term.

Q2: What is the standard form of a parabola?

A2: The standard form of a parabola is given by:

y - k = a(x - h)^2

x - k = a(y - h)^2

where $(h, k)$ is the vertex of the parabola.

Q3: What is the focus of a parabola?

A3: The focus of a parabola is a fixed point that is equidistant from the vertex and the directrix. It is located at a distance of $1/(4a)$ from the vertex.

Q4: What is the directrix of a parabola?

A4: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $1/(4a)$ from the vertex.

Q5: How do I find the focus and directrix of a parabola?

A5: To find the focus and directrix of a parabola, you need to rewrite the equation of the parabola in standard form. Then, use the formula for the focus and directrix to find the desired values.

Q6: What is the vertex of a parabola?

A6: The vertex of a parabola is the point on the parabola that is equidistant from the focus and the directrix. It is the point where the parabola changes direction.

Q7: How do I find the vertex of a parabola?

A7: To find the vertex of a parabola, you need to rewrite the equation of the parabola in standard form. The vertex is then given by the values of $h$ and $k$ in the standard form.

Q8: What is the axis of symmetry of a parabola?

A8: The axis of symmetry of a parabola is a line that passes through the vertex and is perpendicular to the directrix. It is the line that divides the parabola into two equal parts.

Q9: How do I find the axis of symmetry of a parabola?

A9: To find the axis of symmetry of a parabola, you need to rewrite the equation of the parabola in standard form. The axis of symmetry is then given by the line that passes through the vertex and is perpendicular to the directrix.

Q10: What is the difference between a parabola and a circle?

A10: A parabola is a quadratic curve that can be opened upwards or downwards, while a circle is a circular curve that is centered at a fixed point. A parabola has a single axis of symmetry, while a circle has no axis of symmetry.

Conclusion

In this article, we have addressed some of the most frequently asked questions about parabolas. We hope that this article has provided a comprehensive overview of the concepts and formulas related to parabolas. If you have any further questions or doubts, please feel free to ask.

Example Problems

Problem 1

Find the focus and directrix of the parabola:

(x-2)^2=4(y-3)

Solution

We can rewrite this equation as:

x^2 - 4y + 4x + 12 = 0

Completing the square, we get:

(x - 2)^2 = 4(y - 3)

This is the standard form of a parabola, where the vertex is $(2, 3)$ and the equation is:

x - 2 = 4(y - 3)

The focus of the parabola is located at a distance of $1/(4a)$ from the vertex. In this case, $a = 4$ , so the focus is located at:

\left(2 + \frac{1}{16}, 3\right)

y = 3 - \frac{1}{16}

Problem 2

Find the vertex and axis of symmetry of the parabola:

(y-2)^2=6(x-1)

Solution

We can rewrite this equation as:

y^2 - 6x + 6y + 4 = 0

Completing the square, we get:

(y - 2)^2 = 6(x - 1)

This is the standard form of a parabola, where the vertex is $(1, 2)$ and the equation is:

y - 2 = 6(x - 1)

The axis of symmetry of the parabola is a line that passes through the vertex and is perpendicular to the directrix. In this case, the axis of symmetry is a vertical line located at:

x = 1

Final Thoughts

In conclusion, parabolas are an important concept in mathematics, and understanding their properties and formulas is crucial for solving problems in algebra and geometry. We hope that this article has provided a comprehensive overview of the concepts and formulas related to parabolas. If you have any further questions or doubts, please feel free to ask.