Understanding the Basics of a Circle Equation

Oct 13, 2025 by vinooalluri.com 46 views

Which Equation Represents a Circle with a Center at (-4, 9) and a Diameter of 10 Units?

Understanding the Basics of a Circle Equation

A circle is a set of points that are all equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius. The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

Given Information

We are given that the center of the circle is at (-4, 9) and the diameter is 10 units. Since the diameter is twice the radius, the radius is 5 units.

Finding the Correct Equation

To find the correct equation, we need to substitute the values of the center and the radius into the general equation of a circle.

Option A: (x - 9)^2 + (y + 4)^2 = 25

In this option, the center is at (9, -4), which is not the same as the given center (-4, 9). Therefore, this option is incorrect.

Option B: (x + 4)^2 + (y - 9)^2 = 25

In this option, the center is at (-4, 9), which matches the given center. However, the radius squared is 25, which is not equal to the radius squared of 5^2 = 25. This option is also incorrect.

Option C: (x - 9)^2 + (y + 4)^2 = 100

In this option, the center is at (9, -4), which is not the same as the given center (-4, 9). However, the radius squared is 100, which is equal to the radius squared of 5^2 = 25. This option is incorrect.

Option D: (x + 4)^2 + (y - 9)^2 = 100

In this option, the center is at (-4, 9), which matches the given center. The radius squared is 100, which is equal to the radius squared of 5^2 = 25. This option is correct.

Conclusion

The correct equation that represents a circle with a center at (-4, 9) and a diameter of 10 units is (x + 4)^2 + (y - 9)^2 = 100.

Key Takeaways

The equation of a circle can be written in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.
The radius is half the diameter.
To find the correct equation, substitute the values of the center and the radius into the general equation of a circle.

Additional Examples

Find the equation of a circle with a center at (2, 6) and a radius of 3 units.
Find the equation of a circle with a center at (-1, 8) and a diameter of 12 units.

Solving the Examples

To find the equation of a circle with a center at (2, 6) and a radius of 3 units, substitute the values into the general equation of a circle: (x - 2)^2 + (y - 6)^2 = 3^2.
To find the equation of a circle with a center at (-1, 8) and a diameter of 12 units, first find the radius: 12/2 = 6 units. Then, substitute the values into the general equation of a circle: (x + 1)^2 + (y - 8)^2 = 6^2.

Conclusion

In this article, we discussed how to find the equation of a circle with a given center and diameter. We also provided examples and solutions to help readers understand the concept. The correct equation that represents a circle with a center at (-4, 9) and a diameter of 10 units is (x + 4)^2 + (y - 9)^2 = 100.