A Parabola Can Be Represented By The Equation $y^2 = 12x$. Which Equation Represents The Directrix?A. $y = -3$ B. \$y = 3$[/tex\] C. $x = -3$ D. $x = 3$

Introduction

A parabola is a fundamental concept in mathematics, representing a U-shaped curve that opens upwards or downwards. It can be represented by various equations, with the most common form being the quadratic equation in the standard form of $y^2 = 4ax$. However, in this article, we will focus on a parabola represented by the equation $y^2 = 12x$. This equation is a variation of the standard form, where the coefficient of $x$ is not equal to $4a$. Our goal is to find the equation that represents the directrix of this parabola.

Understanding the Parabola Equation

The given equation is $y^2 = 12x$. To understand this equation, let's break it down into its components. The equation is in the form of $y^2 = 4ax$, where $a$ is a constant. In this case, $4a = 12$, which implies that $a = 3$. This means that the parabola opens to the right, as the coefficient of $x$ is positive.

Finding the Focus and Directrix

The focus and directrix of a parabola are two essential components that help us understand its properties. The focus is a fixed point that lies on the axis of symmetry of the parabola, while the directrix is a line that is perpendicular to the axis of symmetry. The focus and directrix are related to each other, as the distance between the focus and the directrix is equal to the distance between the vertex and the focus.

To find the focus and directrix of the given parabola, we need to use the formula for the focus and directrix of a parabola in the form of $y^2 = 4ax$. The focus is located at $(a, 0)$, and the directrix is a line that is perpendicular to the axis of symmetry and is located at a distance of $a$ from the vertex.

Calculating the Focus and Directrix

Using the formula for the focus and directrix, we can calculate the coordinates of the focus and the equation of the directrix. The focus is located at $(a, 0) = (3, 0)$, and the directrix is a line that is perpendicular to the axis of symmetry and is located at a distance of $a$ from the vertex.

Since the parabola opens to the right, the directrix will be a vertical line that is located to the left of the vertex. The equation of the directrix can be found by subtracting $a$ from the x-coordinate of the vertex. Therefore, the equation of the directrix is $x = -3$.

Conclusion

In conclusion, the equation that represents the directrix of the parabola $y^2 = 12x$ is $x = -3$. This equation is derived from the formula for the focus and directrix of a parabola in the form of $y^2 = 4ax$. The focus is located at $(a, 0) = (3, 0)$, and the directrix is a line that is perpendicular to the axis of symmetry and is located at a distance of $a$ from the vertex.

Final Answer

The final answer is $x = -3$.

Introduction

In our previous article, we discussed the equation of a parabola in the form of $y^2 = 12x$ and found the equation that represents the directrix of this parabola. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q1: What is the significance of the directrix in a parabola?

A1: The directrix is a line that is perpendicular to the axis of symmetry of the parabola and is located at a distance of $a$ from the vertex. It plays a crucial role in defining the shape and properties of the parabola.

Q2: How do I find the equation of the directrix of a parabola in the form of $y^2 = 4ax$?

A2: To find the equation of the directrix, you need to use the formula for the focus and directrix of a parabola in the form of $y^2 = 4ax$. The focus is located at $(a, 0)$, and the directrix is a line that is perpendicular to the axis of symmetry and is located at a distance of $a$ from the vertex.

Q3: What is the relationship between the focus and directrix of a parabola?

A3: The focus and directrix of a parabola are related to each other, as the distance between the focus and the directrix is equal to the distance between the vertex and the focus.

Q4: Can you provide an example of a parabola with a directrix?

A4: Consider the parabola $y^2 = 12x$. The equation of the directrix is $x = -3$. This means that the directrix is a vertical line that is located to the left of the vertex.

Q5: How do I determine the orientation of the directrix of a parabola?

A5: To determine the orientation of the directrix, you need to look at the coefficient of $x$ in the equation of the parabola. If the coefficient is positive, the parabola opens to the right, and the directrix will be a vertical line to the left of the vertex. If the coefficient is negative, the parabola opens to the left, and the directrix will be a vertical line to the right of the vertex.

Q6: Can you provide a summary of the key points discussed in this article?

A6: Here is a summary of the key points discussed in this article:

• The equation of a parabola in the form of $y^2 = 12x$ can be represented by the equation $y^2 = 4ax$, where $a = 3$.
• The focus of the parabola is located at $(a, 0) = (3, 0)$.
• The directrix of the parabola is a line that is perpendicular to the axis of symmetry and is located at a distance of $a$ from the vertex.
• The equation of the directrix is $x = -3$.
• The focus and directrix of a parabola are related to each other, as the distance between the focus and the directrix is equal to the distance between the vertex and the focus.

Conclusion

In conclusion, the directrix of a parabola plays a crucial role in defining the shape and properties of the parabola. By understanding the equation of the directrix, you can gain a deeper insight into the properties of the parabola and its applications in various fields.

Final Answer

The final answer is $x = -3$.