How to Find the Equation of a Circle Given Its Center and a Point on Its Circumference

When dealing with circles in mathematics, it is often necessary to find the equation of a circle given its center and a point on its circumference. In this article, we will demonstrate the step-by-step process to find the equation of a circle with the center at -51 and a point on its circumference at (2, -1). This guide will also cover the general equation of a circle and how it relates to the problem at hand.

Understanding the Problem

We are given the center of the circle at (-5, 1) and a point on the circumference at (2, -1). The goal is to find the equation of the circle in standard form.

Deriving the Radius of the Circle

The radius of a circle can be found using the distance formula, which is derived from the Pythagorean theorem. The distance formula for two points, (x1, y1) and (x2, y2), is:

[ r sqrt{(x2 - x1)^2 (y2 - y1)^2} ]

In our case, the center of the circle is (-5, 1) and a point on the circumference is (2, -1). Plugging these values into the formula, we get:

[ r sqrt{(2 - (-5))^2 (-1 - 1)^2} ]

Now, let's simplify this expression:

[ r sqrt{(2 5)^2 (-1 - 1)^2} ] [ r sqrt{7^2 (-2)^2} ] [ r sqrt{49 4} ] [ r sqrt{53} ]

Forming the Equation of the Circle

The general form of the equation of a circle with center (h, k) and radius r is:

[ (x - h)^2 (y - k)^2 r^2 ]

Given that the center is (-5, 1) and the radius is (sqrt{53}), we can substitute these values into the general equation:

[ (x 5)^2 (y - 1)^2 53 ]

Verification

To verify that the equation is correct, we can check if the given point (2, -1) satisfies the equation. Substituting x 2 and y -1 into the equation:

[ (2 5)^2 (-1 - 1)^2 7^2 (-2)^2 ] [ 49 4 53 ]

This confirms that the equation is correct.

In summary, the equation of the circle with center (-5, 1) and a point (2, -1) on its circumference is:

[ (x 5)^2 (y - 1)^2 53 ]

Additional Information

For a deeper understanding of circles and their properties, consider the following:

Center of a Circle: The center of a circle is the fixed point from which all points on the circle are equidistant. Equation of a Circle: The equation of a circle in standard form can be written as ((x - h)^2 (y - k)^2 r^2), where (h, k) is the center and r is the radius. Distance Formula: The distance formula, derived from the Pythagorean theorem, is used to find the distance between two points in a coordinate plane.

By understanding these basic principles, you can solve a wide range of problems related to circles and their equations.

Feel free to explore more problems and concepts related to circles to further enhance your understanding of this fundamental geometric shape.