A Parabola Is Represented By The Equation Y 2 = 5 X Y^2 = 5x Y 2 = 5 X .Which Equation Represents The Directrix?A. Y = − 20 Y = -20 Y = − 20 B. X = − 20 X = -20 X = − 20 C. Y = − 5 4 Y = -\frac{5}{4} Y = − 4 5 ​ D. X = − 5 4 X = -\frac{5}{4} X = − 4 5 ​

A parabola is a fundamental concept in mathematics, representing a U-shaped curve. It is often defined as the set of all points that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix. In this article, we will explore the equation of a parabola and determine the equation of its directrix.

The Equation of a Parabola

The given equation of the parabola is $y^2 = 5x$. This equation represents a parabola that opens to the right, with its vertex at the origin (0, 0). The coefficient of $x$ in the equation is 5, which indicates that the parabola is wider than it is tall.

Understanding Directrices

A directrix is a fixed line that is perpendicular to the axis of symmetry of a parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus. In other words, the directrix is the line that is equidistant from the focus and the parabola.

Finding the Directrix

To find the equation of the directrix, we need to determine the distance from the vertex to the focus. The distance from the vertex to the focus is given by the formula $p = \frac{1}{4a}$, where $a$ is the coefficient of $x$ in the equation of the parabola.

In this case, the coefficient of $x$ is 5, so we have $a = 5$. Plugging this value into the formula, we get $p = \frac{1}{4(5)} = \frac{1}{20}$.

The focus of the parabola is located at a distance $p$ from the vertex, and since the parabola opens to the right, the focus is located at $(p, 0)$. Therefore, the focus of the parabola is located at $\left(\frac{1}{20}, 0\right)$.

Determining the Equation of the Directrix

The directrix is a line that is perpendicular to the axis of symmetry of the parabola. Since the parabola opens to the right, the axis of symmetry is the y-axis. Therefore, the directrix is a horizontal line that is located at a distance $p$ from the vertex.

The equation of the directrix is given by $x = -p$, where $p$ is the distance from the vertex to the focus. Plugging in the value of $p$, we get $x = -\frac{1}{20}$.

However, this is not one of the answer choices. We need to consider the fact that the parabola is represented by the equation $y^2 = 5x$. This equation can be rewritten as $x = \frac{y^2}{5}$. Therefore, the directrix is a line that is located at a distance $p$ from the vertex, and its equation is given by $x = -\frac{1}{20}$.

Conclusion

In conclusion, the equation of the directrix of the parabola represented by the equation $y^2 = 5x$ is $x = -\frac{1}{20}$. This is the correct answer.

Answer

The correct answer is D. $x = -\frac{5}{4}$ is incorrect because it is not the correct equation of the directrix. The correct equation of the directrix is $x = -\frac{1}{20}$.

Why is the answer D. $x = -\frac{5}{4}$ incorrect?

The answer D. $x = -\frac{5}{4}$ is incorrect because it is not the correct equation of the directrix. The correct equation of the directrix is $x = -\frac{1}{20}$.

Why is the answer A. $y = -20$ incorrect?

The answer A. $y = -20$ is incorrect because it is not the correct equation of the directrix. The directrix is a horizontal line that is located at a distance $p$ from the vertex, and its equation is given by $x = -p$. Therefore, the correct equation of the directrix is $x = -\frac{1}{20}$, not $y = -20$.

Why is the answer B. $x = -20$ incorrect?

The answer B. $x = -20$ is incorrect because it is not the correct equation of the directrix. The directrix is a horizontal line that is located at a distance $p$ from the vertex, and its equation is given by $x = -p$. Therefore, the correct equation of the directrix is $x = -\frac{1}{20}$, not $x = -20$.

Why is the answer C. $y = -\frac{5}{4}$ incorrect?

The answer C. $y = -\frac{5}{4}$ is incorrect because it is not the correct equation of the directrix. The directrix is a horizontal line that is located at a distance $p$ from the vertex, and its equation is given by $x = -p$. Therefore, the correct equation of the directrix is $x = -\frac{1}{20}$, not $y = -\frac{5}{4}$.

Final Answer

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In our previous article, we explored the equation of a parabola and determined the equation of its directrix. In this article, we will answer some frequently asked questions about parabolas and directrices.

Q: What is a parabola?

A parabola is a U-shaped curve that is defined by a quadratic equation. It is a fundamental concept in mathematics and has many real-world applications.

A: A parabola is a set of points that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix.

Q: What is the equation of a parabola?

The equation of a parabola is a quadratic equation that can be written in the form $y^2 = 4ax$ or $x^2 = 4ay$. The coefficient of $x$ or $y$ in the equation determines the shape and orientation of the parabola.

A: The equation of a parabola is a quadratic equation that can be written in the form $y^2 = 4ax$ or $x^2 = 4ay$. The coefficient of $x$ or $y$ in the equation determines the shape and orientation of the parabola.

Q: What is the directrix of a parabola?

The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus.

A: The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus.

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

To find the equation of the directrix of a parabola, you need to determine the distance from the vertex to the focus. The distance from the vertex to the focus is given by the formula $p = \frac{1}{4a}$, where $a$ is the coefficient of $x$ or $y$ in the equation of the parabola.

A: To find the equation of the directrix of a parabola, you need to determine the distance from the vertex to the focus. The distance from the vertex to the focus is given by the formula $p = \frac{1}{4a}$, where $a$ is the coefficient of $x$ or $y$ in the equation of the parabola.

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

The focus and the directrix of a parabola are two fixed points that are related to each other. The focus is a point that is equidistant from the directrix and any point on the parabola.

A: The focus and the directrix of a parabola are two fixed points that are related to each other. The focus is a point that is equidistant from the directrix and any point on the parabola.

Q: How do I determine the equation of the directrix of a parabola that opens to the left?

To determine the equation of the directrix of a parabola that opens to the left, you need to use the formula $x = -p$, where $p$ is the distance from the vertex to the focus.

A: To determine the equation of the directrix of a parabola that opens to the left, you need to use the formula $x = -p$, where $p$ is the distance from the vertex to the focus.

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

The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus.

A: The directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus.

Conclusion

In conclusion, the directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus. The equation of the directrix can be determined using the formula $x = -p$, where $p$ is the distance from the vertex to the focus.

Final Answer

The final answer is that the directrix of a parabola is a fixed line that is perpendicular to the axis of symmetry of the parabola. It is the line from which the distance to any point on the parabola is equal to the distance from that point to the focus.