To Find the Point on the Circle

Oct 13, 2025 by recoveryshake.com 94 views

In mathematics, the equation of a circle is a fundamental concept that is used to represent a circle on a coordinate plane. The general equation of a circle is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius of the circle.

The Given Equation

The given equation is:

(x - 3)^2 + (y + 4)^2 = 6^2

This equation represents a circle with a center at (3, -4) and a radius of 6 units.

To Find the Point on the Circle

To find the point on the circle, we need to substitute the x and y values of each option into the equation and check if it satisfies the equation.

Option A: (9, -2)

Substituting x = 9 and y = -2 into the equation:

(9 - 3)^2 + (-2 + 4)^2 = 6^2

Expanding the equation:

6^2 + 2^2 = 6^2

36 + 4 = 36

This equation is true, so option A lies on the circle.

Option B: (0, 11)

Substituting x = 0 and y = 11 into the equation:

(0 - 3)^2 + (11 + 4)^2 = 6^2

Expanding the equation:

(-3)^2 + 15^2 = 6^2

9 + 225 = 36

This equation is not true, so option B does not lie on the circle.

Option C: (3, 10)

Substituting x = 3 and y = 10 into the equation:

(3 - 3)^2 + (10 + 4)^2 = 6^2

Expanding the equation:

0^2 + 14^2 = 6^2

0 + 196 = 36

This equation is not true, so option C does not lie on the circle.

Option D: (-9, 4)

Substituting x = -9 and y = 4 into the equation:

(-9 - 3)^2 + (4 + 4)^2 = 6^2

Expanding the equation:

(-12)^2 + 8^2 = 6^2

144 + 64 = 36

This equation is not true, so option D does not lie on the circle.

Option E: (-3, -4)

Substituting x = -3 and y = -4 into the equation:

(-3 - 3)^2 + (-4 + 4)^2 = 6^2

Expanding the equation:

(-6)^2 + 0^2 = 6^2

36 + 0 = 36

This equation is true, so option E lies on the circle.

Conclusion

Based on the calculations, we can conclude that the points that lie on the circle represented by the equation (x - 3)^2 + (y + 4)^2 = 6^2 are:

Option A: (9, -2)
Option E: (-3, -4)

In this article, we will answer some frequently asked questions about circles, including their equations, properties, and how to find points on a circle.

Q: What is the general equation of a circle?

A: The general equation of a circle is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius of the circle.

Q: What is the center of a circle?

A: The center of a circle is the point (h, k) in the equation of a circle. It is the point from which the radius of the circle is measured.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. It is represented by the variable r in the equation of a circle.

Q: How do I find the points on a circle?

A: To find the points on a circle, you need to substitute the x and y values of each point into the equation of the circle and check if it satisfies the equation.

Q: What is the significance of the radius in a circle?

A: The radius of a circle is important because it determines the size of the circle. A larger radius means a larger circle, while a smaller radius means a smaller circle.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value, representing the distance from the center of the circle to any point on the circle.

Q: Can a circle have a zero radius?

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a single point, not a circle.

Q: How do I graph a circle?

A: To graph a circle, you need to plot the center of the circle and then draw a circle with the given radius. You can use a compass to draw the circle.

Q: What is the relationship between the center and the radius of a circle?

A: The center and the radius of a circle are related in that the center is the point from which the radius is measured. The radius is the distance from the center to any point on the circle.

Q: Can a circle have a negative center?

A: No, a circle cannot have a negative center. The center of a circle is always a point in the coordinate plane, and it cannot have a negative value.

Q: Can a circle have a zero center?

A: No, a circle cannot have a zero center. A circle with a zero center would be a single point, not a circle.

Conclusion

In this article, we have answered some frequently asked questions about circles, including their equations, properties, and how to find points on a circle. We hope that this article has been helpful in understanding the basics of circles.