Understanding Hyperbola Shift

Oct 13, 2025 by windailys.com 14 views

Understanding Hyperbola Shift

A hyperbola is a type of conic section that can be shifted horizontally and vertically. When a hyperbola is shifted, its center changes. In this article, we will explore how to calculate the new center of a hyperbola after it has been shifted to the right by 4 units and upward by 3 units.

Given Hyperbola Equation

The given hyperbola equation is:

\frac{(x-5)^2}{7}-\frac{(y+3)^2}{9}=1

This equation represents a hyperbola with a center at (5, -3).

Shifting the Hyperbola

To shift the hyperbola to the right by 4 units and upward by 3 units, we need to adjust the x and y coordinates of the center.

Shifting the Hyperbola to the Right

When shifting the hyperbola to the right by 4 units, the new x-coordinate of the center will be:

x_{new} = x_{old} + 4

Shifting the Hyperbola Upward

When shifting the hyperbola upward by 3 units, the new y-coordinate of the center will be:

y_{new} = y_{old} + 3

Calculating the New Center

Using the given hyperbola equation, we can calculate the new center of the hyperbola.

New x-Coordinate

The new x-coordinate of the center is:

x_{new} = 5 + 4 = 9

New y-Coordinate

The new y-coordinate of the center is:

y_{new} = -3 + 3 = 0

New Center of the Hyperbola

The new center of the hyperbola is (9, 0).

Conclusion

In this article, we have explored how to calculate the new center of a hyperbola after it has been shifted to the right by 4 units and upward by 3 units. We have used the given hyperbola equation to calculate the new x and y coordinates of the center. The new center of the hyperbola is (9, 0).

Key Takeaways

A hyperbola can be shifted horizontally and vertically.
The new center of a hyperbola can be calculated by adjusting the x and y coordinates of the original center.
The new x-coordinate of the center is calculated by adding the shift amount to the original x-coordinate.
The new y-coordinate of the center is calculated by adding the shift amount to the original y-coordinate.

Practice Problems

A hyperbola is shifted to the left by 2 units and downward by 1 unit. The original center of the hyperbola is (3, 2). What is the new center of the hyperbola?
A hyperbola is shifted to the right by 3 units and upward by 2 units. The original center of the hyperbola is (-2, -1). What is the new center of the hyperbola?

Answer Key

The new center of the hyperbola is (1, 1).
The new center of the hyperbola is (-1, 1).
Hyperbola Shift and Center Calculation: Q&A =====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about hyperbola shift and center calculation.

Q: What is a hyperbola?

A: A hyperbola is a type of conic section that can be shifted horizontally and vertically. It is a set of points that are equidistant from two fixed points called foci.

Q: How do I calculate the new center of a hyperbola after it has been shifted?

A: To calculate the new center of a hyperbola after it has been shifted, you need to adjust the x and y coordinates of the original center. The new x-coordinate is calculated by adding the shift amount to the original x-coordinate, and the new y-coordinate is calculated by adding the shift amount to the original y-coordinate.

Q: What is the formula for calculating the new center of a hyperbola?

A: The formula for calculating the new center of a hyperbola is:

x_{new} = x_{old} + \Delta x

y_{new} = y_{old} + \Delta y

where $x_{old}$ and $y_{old}$ are the original x and y coordinates of the center, and $\Delta x$ and $\Delta y$ are the shift amounts.

Q: How do I determine the shift amounts?

A: The shift amounts are the amounts by which the hyperbola is shifted horizontally and vertically. For example, if the hyperbola is shifted to the right by 4 units and upward by 3 units, the shift amounts are $\Delta x = 4$ and $\Delta y = 3$ .

Q: Can I shift a hyperbola in any direction?

A: Yes, you can shift a hyperbola in any direction. However, the shift amounts must be specified in terms of the x and y coordinates of the center.

Q: How do I apply the shift amounts to the original center?

A: To apply the shift amounts to the original center, you simply add the shift amounts to the original x and y coordinates of the center.

Q: What is the new center of the hyperbola in the example given in the article?

A: The new center of the hyperbola is (9, 0).

Q: Can I use the same formula to calculate the new center of a hyperbola that has been rotated?

A: No, the formula for calculating the new center of a hyperbola that has been rotated is different from the formula for calculating the new center of a hyperbola that has been shifted. The formula for calculating the new center of a hyperbola that has been rotated is:

x_{new} = x_{old} \cos \theta - y_{old} \sin \theta

y_{new} = x_{old} \sin \theta + y_{old} \cos \theta

where $\theta$ is the angle of rotation.

Conclusion

In this article, we have answered some of the most frequently asked questions about hyperbola shift and center calculation. We have provided formulas and examples to help you understand how to calculate the new center of a hyperbola after it has been shifted.

Key Takeaways

A hyperbola can be shifted horizontally and vertically.
The new center of a hyperbola can be calculated by adjusting the x and y coordinates of the original center.
The new x-coordinate of the center is calculated by adding the shift amount to the original x-coordinate.
The new y-coordinate of the center is calculated by adding the shift amount to the original y-coordinate.
The formula for calculating the new center of a hyperbola that has been rotated is different from the formula for calculating the new center of a hyperbola that has been shifted.

Practice Problems

A hyperbola is shifted to the left by 2 units and downward by 1 unit. The original center of the hyperbola is (3, 2). What is the new center of the hyperbola?
A hyperbola is shifted to the right by 3 units and upward by 2 units. The original center of the hyperbola is (-2, -1). What is the new center of the hyperbola?
A hyperbola is rotated by 30 degrees counterclockwise. The original center of the hyperbola is (1, 2). What is the new center of the hyperbola?

Answer Key

The new center of the hyperbola is (1, 1).
The new center of the hyperbola is (-1, 1).
The new center of the hyperbola is (1.5, 1.5).