Effect of Shifting on Circle Equation

Oct 13, 2025 by windailys.com 129 views

Understanding Circle Equations

A circle is a set of points in a plane that are equidistant from a central point known as the center. The general equation of a circle with center (h, k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2. In this case, the equation of the circle is (x-3)^2 + (y-5)^2 = 36. This means that the center of the circle is at (3, 5) and the radius is 6 units.

Shifting a Circle

Shifting a circle means moving it from its original position to a new position. When a circle is shifted down 2 units, its center is moved down by 2 units. This is because the y-coordinate of the center of the circle is decreased by 2 units.

Effect of Shifting on Circle Equation

When a circle is shifted down 2 units, its equation changes. The new equation of the circle can be obtained by replacing the y-coordinate of the center with the new y-coordinate. In this case, the new y-coordinate is 5 - 2 = 3. Therefore, the new equation of the circle is (x-3)^2 + (y-3)^2 = 36.

Analyzing the Options

Now, let's analyze the options given in the problem.

Option A: Both the x- and y-coordinates of the center of the circle increase by 2.

This option is incorrect because the y-coordinate of the center of the circle decreases by 2 units, not increases.

Option B: Both the x- and y-coordinates of the center of the circle decrease by 2.

This option is also incorrect because the x-coordinate of the center of the circle remains the same, and only the y-coordinate decreases by 2 units.

Option C: The y-coordinate of the center of the circle decreases by 2 units.

This option is correct because the y-coordinate of the center of the circle decreases by 2 units when the circle is shifted down 2 units.

Conclusion

In conclusion, when a circle with the equation (x-3)^2 + (y-5)^2 = 36 is shifted down 2 units, the y-coordinate of the center of the circle decreases by 2 units. Therefore, the correct option is C.

Example Problems

Here are a few example problems to help you understand the concept better.

Example 1

A circle with the equation (x-2)^2 + (y-4)^2 = 16 is shifted up 3 units. What is the new equation of the circle?

Solution

When the circle is shifted up 3 units, the y-coordinate of the center of the circle increases by 3 units. Therefore, the new y-coordinate is 4 + 3 = 7. The new equation of the circle is (x-2)^2 + (y-7)^2 = 16.

Example 2

A circle with the equation (x-5)^2 + (y-2)^2 = 9 is shifted down 4 units. What is the new equation of the circle?

Solution

When the circle is shifted down 4 units, the y-coordinate of the center of the circle decreases by 4 units. Therefore, the new y-coordinate is 2 - 4 = -2. The new equation of the circle is (x-5)^2 + (y+2)^2 = 9.

Practice Problems

Here are a few practice problems to help you practice the concept.

Practice Problem 1

A circle with the equation (x-1)^2 + (y-6)^2 = 25 is shifted up 2 units. What is the new equation of the circle?

Practice Problem 2

A circle with the equation (x-4)^2 + (y-8)^2 = 4 is shifted down 3 units. What is the new equation of the circle?

Practice Problem 3

A circle with the equation (x-2)^2 + (y-3)^2 = 9 is shifted left 4 units. What is the new equation of the circle?

Practice Problem 4

A circle with the equation (x-6)^2 + (y-1)^2 = 16 is shifted right 2 units. What is the new equation of the circle?

Practice Problem 5

A circle with the equation (x-3)^2 + (y-5)^2 = 36 is shifted up 4 units. What is the new equation of the circle?

Solutions to Practice Problems

Here are the solutions to the practice problems.

Solution to Practice Problem 1

When the circle is shifted up 2 units, the y-coordinate of the center of the circle increases by 2 units. Therefore, the new y-coordinate is 6 + 2 = 8. The new equation of the circle is (x-1)^2 + (y-8)^2 = 25.

Solution to Practice Problem 2

When the circle is shifted down 3 units, the y-coordinate of the center of the circle decreases by 3 units. Therefore, the new y-coordinate is 8 - 3 = 5. The new equation of the circle is (x-4)^2 + (y-5)^2 = 4.

Solution to Practice Problem 3

When the circle is shifted left 4 units, the x-coordinate of the center of the circle decreases by 4 units. Therefore, the new x-coordinate is 2 - 4 = -2. The new equation of the circle is (x+2)^2 + (y-3)^2 = 9.

Solution to Practice Problem 4

When the circle is shifted right 2 units, the x-coordinate of the center of the circle increases by 2 units. Therefore, the new x-coordinate is 6 + 2 = 8. The new equation of the circle is (x-8)^2 + (y-1)^2 = 16.

Solution to Practice Problem 5

When the circle is shifted up 4 units, the y-coordinate of the center of the circle increases by 4 units. Therefore, the new y-coordinate is 5 + 4 = 9. The new equation of the circle is (x-3)^2 + (y-9)^2 = 36.

Conclusion

In conclusion, shifting a circle means moving it from its original position to a new position. When a circle is shifted down 2 units, its center is moved down by 2 units. The new equation of the circle can be obtained by replacing the y-coordinate of the center with the new y-coordinate. Therefore, the correct option is C.

Understanding Circle Equations

Shifting a Circle

Q&A

Q: What is the effect of shifting a circle on its equation?

A: When a circle is shifted, its equation changes. The new equation of the circle can be obtained by replacing the y-coordinate of the center with the new y-coordinate.

Q: How do I find the new equation of a circle after it has been shifted?

A: To find the new equation of a circle after it has been shifted, you need to replace the y-coordinate of the center with the new y-coordinate. For example, if the original equation of the circle is (x-3)^2 + (y-5)^2 = 36 and it is shifted down 2 units, the new equation of the circle is (x-3)^2 + (y-3)^2 = 36.

Q: What is the difference between shifting a circle up and shifting a circle down?

A: Shifting a circle up means moving it from its original position to a new position where its center is moved up. Shifting a circle down means moving it from its original position to a new position where its center is moved down.

Q: Can a circle be shifted in both the x and y directions?

A: Yes, a circle can be shifted in both the x and y directions. For example, if a circle is shifted 2 units to the right and 3 units up, its center will be moved 2 units to the right and 3 units up.

Q: How do I find the new equation of a circle after it has been shifted in both the x and y directions?

A: To find the new equation of a circle after it has been shifted in both the x and y directions, you need to replace the x-coordinate and y-coordinate of the center with the new x-coordinate and y-coordinate. For example, if the original equation of the circle is (x-3)^2 + (y-5)^2 = 36 and it is shifted 2 units to the right and 3 units up, the new equation of the circle is (x-5)^2 + (y-8)^2 = 36.

Q: Can a circle be shifted in only one direction?

A: Yes, a circle can be shifted in only one direction. For example, if a circle is shifted 2 units to the right, its center will be moved 2 units to the right.

Q: How do I find the new equation of a circle after it has been shifted in only one direction?

A: To find the new equation of a circle after it has been shifted in only one direction, you need to replace the x-coordinate or y-coordinate of the center with the new x-coordinate or y-coordinate. For example, if the original equation of the circle is (x-3)^2 + (y-5)^2 = 36 and it is shifted 2 units to the right, the new equation of the circle is (x-5)^2 + (y-5)^2 = 36.

Conclusion

In conclusion, shifting a circle means moving it from its original position to a new position. When a circle is shifted, its equation changes. The new equation of the circle can be obtained by replacing the x-coordinate and y-coordinate of the center with the new x-coordinate and y-coordinate. Therefore, it is essential to understand how to find the new equation of a circle after it has been shifted.

Example Problems

Here are a few example problems to help you understand the concept better.

Example 1

A circle with the equation (x-2)^2 + (y-4)^2 = 16 is shifted up 3 units. What is the new equation of the circle?

Solution

Example 2

A circle with the equation (x-5)^2 + (y-2)^2 = 9 is shifted down 4 units. What is the new equation of the circle?

Solution

Practice Problems

Here are a few practice problems to help you practice the concept.

Practice Problem 1

A circle with the equation (x-1)^2 + (y-6)^2 = 25 is shifted up 2 units. What is the new equation of the circle?

Practice Problem 2

A circle with the equation (x-4)^2 + (y-8)^2 = 4 is shifted down 3 units. What is the new equation of the circle?

Practice Problem 3

A circle with the equation (x-2)^2 + (y-3)^2 = 9 is shifted left 4 units. What is the new equation of the circle?

Practice Problem 4

A circle with the equation (x-6)^2 + (y-1)^2 = 16 is shifted right 2 units. What is the new equation of the circle?

Practice Problem 5

A circle with the equation (x-3)^2 + (y-5)^2 = 36 is shifted up 4 units. What is the new equation of the circle?