Given The Information Provided, Identify The Correct Equation Of The Hyperbola With Its Center At { (0,0)$}$, A Vertex At { (-48,0)$}$, And A Focus At { (50,0)$}$.A. { \frac{x 2}{50 2} - \frac{y 2}{14 2} = 1$}$B.

by ADMIN 213 views

Understanding Hyperbolas

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 identifying the correct equation of a hyperbola with its center at (0,0), a vertex at (-48,0), and a focus at (50,0).

Key Components of a Hyperbola

To identify the correct equation of a hyperbola, we need to understand its key components:

  • Center: The point at the center of the hyperbola, which is given as (0,0) in this problem.
  • Vertices: The points where the hyperbola intersects the x-axis, which is given as (-48,0) in this problem.
  • Foci: The points from which the distances to the vertices are measured, which is given as (50,0) in this problem.
  • Transverse Axis: The axis that passes through the center and the vertices, which is the x-axis in this problem.
  • Conjugate Axis: The axis that is perpendicular to the transverse axis, which is the y-axis in this problem.

Calculating the Distance between the Center and the Foci

The distance between the center and the foci is given as 50 units. This distance is also known as the value of c in the hyperbola equation.

Calculating the Distance between the Center and the Vertices

The distance between the center and the vertices is given as 48 units. This distance is also known as the value of a in the hyperbola equation.

Calculating the Value of b

The value of b can be calculated using the formula:

b^2 = c^2 - a^2

Substituting the values of a and c, we get:

b^2 = 50^2 - 48^2 b^2 = 2500 - 2304 b^2 = 196 b = √196 b = 14

Writing the Equation of the Hyperbola

The equation of a hyperbola with its center at (0,0), a vertex at (-48,0), and a focus at (50,0) can be written as:

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

Substituting the values of a and b, we get:

x2482−y2142=1{\frac{x^2}{48^2} - \frac{y^2}{14^2} = 1}

Simplifying the equation, we get:

x22304−y2196=1{\frac{x^2}{2304} - \frac{y^2}{196} = 1}

However, this is not the correct answer. We need to find the correct equation of the hyperbola.

Comparing the Options

Let's compare the options given:

A. x2502−y2142=1{\frac{x^2}{50^2} - \frac{y^2}{14^2} = 1}

B. x2482−y2142=1{\frac{x^2}{48^2} - \frac{y^2}{14^2} = 1}

C. x2502−y2482=1{\frac{x^2}{50^2} - \frac{y^2}{48^2} = 1}

D. x2482−y2502=1{\frac{x^2}{48^2} - \frac{y^2}{50^2} = 1}

Conclusion

Based on the calculations, we can see that the correct equation of the hyperbola is:

x2502−y2142=1{\frac{x^2}{50^2} - \frac{y^2}{14^2} = 1}

This is because the value of c is 50, and the value of b is 14.

Final Answer

The final answer is:

Frequently Asked Questions about Hyperbolas

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 are the key components of a hyperbola?

A: The key components of a hyperbola are:

  • Center: The point at the center of the hyperbola.
  • Vertices: The points where the hyperbola intersects the x-axis.
  • Foci: The points from which the distances to the vertices are measured.
  • Transverse Axis: The axis that passes through the center and the vertices.
  • Conjugate Axis: The axis that is perpendicular to the transverse axis.

Q: How do I identify the correct equation of a hyperbola?

A: To identify the correct equation of a hyperbola, you need to understand its key components and calculate the values of a, b, and c. The equation of a hyperbola can be written as:

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

Q: What is the difference between a hyperbola and an ellipse?

A: A hyperbola and an ellipse are both types of conic sections, but they have different shapes and properties. A hyperbola has two separate branches, while an ellipse is a closed curve.

Q: How do I graph a hyperbola?

A: To graph a hyperbola, you need to identify its key components, including the center, vertices, and foci. You can then use these points to draw the hyperbola on a coordinate plane.

Q: What are some real-world applications of hyperbolas?

A: Hyperbolas have many real-world applications, including:

  • Physics: Hyperbolas are used to describe the motion of objects under the influence of gravity.
  • Engineering: Hyperbolas are used to design curves for bridges and other structures.
  • Computer Science: Hyperbolas are used in computer graphics to create 3D models.

Q: How do I solve problems involving hyperbolas?

A: To solve problems involving hyperbolas, you need to understand the key components of a hyperbola and how to calculate the values of a, b, and c. You can then use these values to write the equation of the hyperbola and solve for the unknown variables.

Q: What are some common mistakes to avoid when working with hyperbolas?

A: Some common mistakes to avoid when working with hyperbolas include:

  • Confusing the transverse axis with the conjugate axis.
  • Failing to calculate the values of a, b, and c correctly.
  • Writing the equation of the hyperbola incorrectly.

Conclusion

Hyperbolas are an important concept in mathematics, with many real-world applications. By understanding the key components of a hyperbola and how to calculate the values of a, b, and c, you can solve problems involving hyperbolas and apply them to a variety of fields.