A Hyperbola Centered At The Origin Has A Vertex At ( − 6 , 0 (-6,0 ( − 6 , 0 ] And A Focus At ( 10 , 0 (10,0 ( 10 , 0 ]. Which Are The Equations Of The Directrices?A. X = ± 18 5 X = \pm \frac{18}{5} X = ± 5 18 ​ B. Y = ± 18 5 Y = \pm \frac{18}{5} Y = ± 5 18 ​ C. Y = ± 3 5 Y = \pm \frac{3}{5} Y = ± 5 3 ​ D.

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Introduction

A hyperbola is a type of mathematical curve that consists of two separate branches. It is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (called foci) is constant. In this article, we will focus on a hyperbola centered at the origin with a vertex at (-6,0) and a focus at (10,0). We will explore the equations of the directrices, which are lines that are perpendicular to the transverse axis and pass through the foci.

Understanding the Hyperbola

A hyperbola can be defined in two ways: in terms of its vertices and foci, or in terms of its asymptotes. The standard form of a hyperbola centered at the origin is given by:

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

where aa and bb are the distances from the center to the vertices and co-vertices, respectively. The foci of the hyperbola are located at (±c,0)(\pm c, 0), where c2=a2+b2c^2 = a^2 + b^2.

Finding the Distance from the Center to the Vertex

The vertex of the hyperbola is located at (-6,0), which means that the distance from the center to the vertex is a=6a = 6. This is because the vertex is on the x-axis, and the center is at the origin.

Finding the Distance from the Center to the Focus

The focus of the hyperbola is located at (10,0), which means that the distance from the center to the focus is c=10c = 10. This is because the focus is on the x-axis, and the center is at the origin.

Finding the Value of b

We can use the formula c2=a2+b2c^2 = a^2 + b^2 to find the value of bb. Plugging in the values we know, we get:

102=62+b210^2 = 6^2 + b^2

Simplifying, we get:

100=36+b2100 = 36 + b^2

Subtracting 36 from both sides, we get:

64=b264 = b^2

Taking the square root of both sides, we get:

b=±64b = \pm \sqrt{64}

b=±8b = \pm 8

Finding the Equations of the Directrices

The directrices of a hyperbola are lines that are perpendicular to the transverse axis and pass through the foci. The equation of the directrices can be found using the formula:

x=±a2cx = \pm \frac{a^2}{c}

Plugging in the values we know, we get:

x=±6210x = \pm \frac{6^2}{10}

Simplifying, we get:

x=±3610x = \pm \frac{36}{10}

x=±185x = \pm \frac{18}{5}

Conclusion

In this article, we have explored the equations of the directrices of a hyperbola centered at the origin with a vertex at (-6,0) and a focus at (10,0). We have found that the equations of the directrices are x=±185x = \pm \frac{18}{5}. This is because the directrices are lines that are perpendicular to the transverse axis and pass through the foci, and the equation of the directrices can be found using the formula x=±a2cx = \pm \frac{a^2}{c}.

Discussion

The equations of the directrices of a hyperbola are important because they help us understand the shape and properties of the hyperbola. The directrices are lines that are perpendicular to the transverse axis and pass through the foci, and they play a crucial role in defining the shape of the hyperbola. In this article, we have seen how to find the equations of the directrices using the formula x=±a2cx = \pm \frac{a^2}{c}.

Final Answer

The final answer is: A\boxed{A}

Introduction

In our previous article, we explored the equations of the directrices of a hyperbola centered at the origin with a vertex at (-6,0) and a focus at (10,0). We found that the equations of the directrices are x=±185x = \pm \frac{18}{5}. In this article, we will answer some frequently asked questions about hyperbolas and their directrices.

Q: What is a hyperbola?

A: A hyperbola is a type of mathematical curve that consists of two separate branches. It is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (called foci) is constant.

Q: What is the standard form of a hyperbola centered at the origin?

A: The standard form of a hyperbola centered at the origin is given by:

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

where aa and bb are the distances from the center to the vertices and co-vertices, respectively.

Q: How do I find the distance from the center to the vertex?

A: To find the distance from the center to the vertex, you need to know the coordinates of the vertex. In this case, the vertex is located at (-6,0), which means that the distance from the center to the vertex is a=6a = 6.

Q: How do I find the distance from the center to the focus?

A: To find the distance from the center to the focus, you need to know the coordinates of the focus. In this case, the focus is located at (10,0), which means that the distance from the center to the focus is c=10c = 10.

Q: How do I find the value of b?

A: To find the value of bb, you can use the formula c2=a2+b2c^2 = a^2 + b^2. Plugging in the values we know, we get:

102=62+b210^2 = 6^2 + b^2

Simplifying, we get:

100=36+b2100 = 36 + b^2

Subtracting 36 from both sides, we get:

64=b264 = b^2

Taking the square root of both sides, we get:

b=±64b = \pm \sqrt{64}

b=±8b = \pm 8

Q: How do I find the equations of the directrices?

A: To find the equations of the directrices, you can use the formula:

x=±a2cx = \pm \frac{a^2}{c}

Plugging in the values we know, we get:

x=±6210x = \pm \frac{6^2}{10}

Simplifying, we get:

x=±3610x = \pm \frac{36}{10}

x=±185x = \pm \frac{18}{5}

Q: What is the significance of the directrices?

A: The directrices are lines that are perpendicular to the transverse axis and pass through the foci. They play a crucial role in defining the shape of the hyperbola.

Q: Can you give an example of a hyperbola with a different orientation?

A: Yes, a hyperbola can be oriented in different ways. For example, a hyperbola with a vertex at (0,6) and a focus at (0,10) would have a different equation.

Q: How do I graph a hyperbola?

A: To graph a hyperbola, you need to know the equation of the hyperbola. You can use the equation to find the coordinates of the vertices and foci, and then plot the points on a coordinate plane.

Q: Can you give a real-world example of a hyperbola?

A: Yes, a hyperbola can be used to model the path of a projectile under the influence of gravity. For example, the trajectory of a thrown ball can be modeled using a hyperbola.

Conclusion

In this article, we have answered some frequently asked questions about hyperbolas and their directrices. We have seen how to find the distance from the center to the vertex, the distance from the center to the focus, and the value of bb. We have also seen how to find the equations of the directrices and the significance of the directrices.