A Parabola Can Be Represented By The Equation Y 2 = 12 X Y^2 = 12x Y 2 = 12 X .Which Equation Represents The Directrix?A. Y = − 3 Y = -3 Y = − 3 B. Y = 3 Y = 3 Y = 3 C. X = − 3 X = -3 X = − 3 D. X = 3 X = 3 X = 3

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Introduction

A parabola is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. The equation $y^2 = 12x$ represents a parabola that opens to the right. In this article, we will explore the concept of a directrix and determine which equation represents the directrix of the given parabola.

What is a Parabola?

A parabola is a quadratic curve that can be represented by the equation $y^2 = 4ax$ or $x^2 = 4ay$, where $a$ is a constant. The parabola opens to the right if $a$ is positive and to the left if $a$ is negative. The parabola opens upwards if $a$ is positive and downwards if $a$ is negative.

What is a Directrix?

A directrix is a line that is perpendicular to the axis of symmetry of a parabola. It is a line that is not part of the parabola but is used to define its shape. The directrix is a key concept in the study of parabolas, and it is used to determine the equation of a parabola.

Finding the Directrix of a Parabola

To find the directrix of a parabola, we need to use the equation of the parabola and the definition of a directrix. The equation of a parabola is given by $y^2 = 4ax$, where $a$ is a constant. 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 $a$ units from the vertex of the parabola.

Finding the Equation of the Directrix

To find the equation of the directrix, we need to use the equation of the parabola and the definition of a directrix. The equation of the parabola is given by $y^2 = 12x$. We can rewrite this equation as $y^2 = 4(3)x$. Comparing this equation with the standard equation of a parabola, we can see that $a = 3$. Therefore, the directrix of the parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $3$ units from the vertex of the parabola.

Determining the Equation of the Directrix

Since the parabola opens to the right, the directrix is a vertical line. The equation of a vertical line is given by $x = c$, where $c$ is a constant. Since the directrix is located at a distance of $3$ units from the vertex of the parabola, the equation of the directrix is $x = 3$.

Conclusion

In conclusion, the equation of the directrix of the parabola $y^2 = 12x$ is $x = 3$. This equation represents a vertical line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $3$ units from the vertex of the parabola.

Frequently Asked Questions

• What is a parabola? A parabola is a quadratic curve that can be represented by the equation $y^2 = 4ax$ or $x^2 = 4ay$, where $a$ is a constant.
• What is a directrix? A directrix is a line that is perpendicular to the axis of symmetry of a parabola.
• How do you find the equation of the directrix of a parabola? To find the equation of the directrix of a parabola, you need to use the equation of the parabola and the definition of a directrix.

References

• [1] "Parabola" by Math Open Reference. Math Open Reference. https://www.mathopenref.com/parabola.html
• [2] "Directrix" by Math Is Fun. Math Is Fun. https://www.mathisfun.com/geometry/directrix.html

Final Answer

The final answer is: $\boxed{D. x = 3}$

$$

Introduction

A parabola is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. The equation $y^2 = 12x$ represents a parabola that opens to the right. In this article, we will explore the concept of a directrix and determine which equation represents the directrix of the given parabola.

What is a Parabola?

A parabola is a quadratic curve that can be represented by the equation $y^2 = 4ax$ or $x^2 = 4ay$, where $a$ is a constant. The parabola opens to the right if $a$ is positive and to the left if $a$ is negative. The parabola opens upwards if $a$ is positive and downwards if $a$ is negative.

What is a Directrix?

A directrix is a line that is perpendicular to the axis of symmetry of a parabola. It is a line that is not part of the parabola but is used to define its shape. The directrix is a key concept in the study of parabolas, and it is used to determine the equation of a parabola.

Finding the Directrix of a Parabola

To find the directrix of a parabola, we need to use the equation of the parabola and the definition of a directrix. The equation of a parabola is given by $y^2 = 4ax$, where $a$ is a constant. 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 $a$ units from the vertex of the parabola.

Finding the Equation of the Directrix

To find the equation of the directrix, we need to use the equation of the parabola and the definition of a directrix. The equation of the parabola is given by $y^2 = 12x$. We can rewrite this equation as $y^2 = 4(3)x$. Comparing this equation with the standard equation of a parabola, we can see that $a = 3$. Therefore, the directrix of the parabola is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $3$ units from the vertex of the parabola.

Determining the Equation of the Directrix

Since the parabola opens to the right, the directrix is a vertical line. The equation of a vertical line is given by $x = c$, where $c$ is a constant. Since the directrix is located at a distance of $3$ units from the vertex of the parabola, the equation of the directrix is $x = 3$.

Conclusion

In conclusion, the equation of the directrix of the parabola $y^2 = 12x$ is $x = 3$. This equation represents a vertical line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $3$ units from the vertex of the parabola.

Frequently Asked Questions

Q: What is a parabola?

A: A parabola is a quadratic curve that can be represented by the equation $y^2 = 4ax$ or $x^2 = 4ay$, where $a$ is a constant.

Q: What is a directrix?

A: A directrix is a line that is perpendicular to the axis of symmetry of a parabola.

Q: How do you find the equation of the directrix of a parabola?

A: To find the equation of the directrix of a parabola, you need to use the equation of the parabola and the definition of a directrix.

Q: What is the equation of the directrix of the parabola $y^2 = 12x$?

A: The equation of the directrix of the parabola $y^2 = 12x$ is $x = 3$.

Q: What is the definition of a parabola?

A: A parabola is a quadratic curve that can be represented by the equation $y^2 = 4ax$ or $x^2 = 4ay$, where $a$ is a constant.

Q: What is the definition of a directrix?

A: A directrix is a line that is perpendicular to the axis of symmetry of a parabola.

Q: How do you determine the equation of the directrix of a parabola?

A: To determine the equation of the directrix of a parabola, you need to use the equation of the parabola and the definition of a directrix.

Additional Questions and Answers

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the parabola that is equidistant from the focus and the directrix.

Q: What is the focus of a parabola?

A: The focus of a parabola is the point on the axis of symmetry of the parabola that is equidistant from the vertex and the directrix.

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

A: To find the vertex of a parabola, you need to use the equation of the parabola and the definition of a vertex.

Q: How do you find the focus of a parabola?

A: To find the focus of a parabola, you need to use the equation of the parabola and the definition of a focus.

Conclusion

In conclusion, the equation of the directrix of the parabola $y^2 = 12x$ is $x = 3$. This equation represents a vertical line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $3$ units from the vertex of the parabola. We hope that this article has provided you with a better understanding of the concept of a directrix and how to determine the equation of a directrix of a parabola.

Final Answer

The final answer is: $\boxed{D. x = 3}$