91Ó°ÊÓ

To find the radius of a circle, we use the distance formula. This formula helps us calculate the distance between two points in a coordinate plane. In the context of a circle, the formula can determine the distance between the circle's center and any other point on its circumference. For this exercise, the circle is centered at \((-1, 5)\) and it passes through the point \((-4, -6)\).Let’s apply the distance formula: \[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the points,\[r = \sqrt{((-4 - (-1))^2 + (-6 - 5)^2)}\]Which simplifies to \[r = \sqrt{((-3)^2 + (-11)^2)}\]This gives us \[r = \sqrt{9 + 121} = \sqrt{130}\].The radius, therefore, is the square root of 130. This length forms a straight line from the center to any point on the circle, making it a key component of the circle's equation.

The center of a circle is a fixed point in the plane from which all points on the circle are at a constant distance (the radius). In this situation, the center is provided as \((-1, 5)\).

• The x-coordinate of the center is \(h = -1\)
• The y-coordinate is \(k = 5\)

The role of the center in the circle equation is crucial because it becomes part of the standard form equation of the circle.Knowing the center helps us set the position of the circle on the coordinate plane, and influences the equation's symmetry. It aids in determining the behavior of the circle with respect to the x and y axes.When you graph a circle, the center is where you start, and you pave your way outwards with the radius, providing a full overview of the circle's reach across the plane.

At the heart of solving circle problems lies the general equation of a circle. The equation is expressed as:\[(x - h)^2 + (y - k)^2 = r^2\]Where \((h, k)\) denotes the center of the circle, and \(r\) is the radius.To make this equation actionable, you always need:

• The center coordinates, \((h, k)\)
• The circle's radius, \(r\)

In our exercise, substituting \(h = -1\) and \(k = 5\), the equation becomes:\[(x + 1)^2 + (y - 5)^2 = r^2\]Once the radius is calculated or given, you substitute \(r^2 = 130\). Thus, the finalized equation is:\[(x + 1)^2 + (y - 5)^2 = 130\].This equation articulates the relationship between any point \((x, y)\)on the circle and its defined center and radius. It's the blueprint for constructing a circle within a coordinate system.