Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((2,-7) ;\) passing through \((-2,-4)\)

Step-by-Step Solution

Answer

The center-radius form is \((x - 2)^2 + (y + 7)^2 = 25\).

Step 1: Identify the formula needed

To write the equation of a circle, we use the center-radius form which 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. In this problem, we already have the center \((2, -7)\).

Step 2: Calculate the radius

The radius \(r\) is the distance from the center \((2, -7)\) to the point \((-2, -4)\), which lies on the circle. Use the distance formula: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting the values, we have: \[r = \sqrt{(-2 - 2)^2 + (-4 + 7)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\]

Step 3: Write the equation in center-radius form

With the center \((2, -7)\) and the radius \(r = 5\), substitute these values into the center-radius form of a circle's equation: \((x - 2)^2 + (y + 7)^2 = 5^2\). Simplified, this becomes \((x - 2)^2 + (y + 7)^2 = 25\).

Key Concepts

Center-radius form

The center-radius form of a circle is a specific way of writing the equation of a circle. It is incredibly useful in geometry and algebra. This form provides a clear connection between the algebraic representation and the geometric features of a circle. The center-radius form is given by the equation

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius.
The left side of the equation indicates the squared distances from any point \((x, y)\) on the circle to the center \((h, k)\). The equation ensures that this distance is equal to the radius \(r\).
This form is especially handy when you know the center and radius or need to derive them from a given circle equation.

Distance formula

Calculating the radius of a circle requires finding the distance between two points: the center and a point on the circle. This process uses the distance formula, a crucial tool in coordinate geometry.
The distance formula is written as:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula calculates the distance \(d\) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane.
In the context of circles, this distance corresponds to the radius when one point is the circle's center and the other point is on its perimeter. Taking these steps ensures that the calculated radius is accurate.

Circle center

The center of a circle is an essential concept in geometry. It is the point equidistant from all points on the circle's edge.
In the context of the center-radius form, the center of the circle is denoted by the point \((h, k)\).
This notation indicates the horizontal and vertical distances from the origin to the center of the circle on the plane. Knowing the center is crucial as it helps lay the foundation for calculating the radius and forming the equation of the circle.

• The coordinate \(h\) represents the horizontal distance from the origin.
• The coordinate \(k\) signifies the vertical distance.

By identifying these coordinates, we can better understand and visualize the circle's placement in space.

Radius calculation

Radius calculation involves using the distance formula to find the length of the radius from the center of the circle to any point on its perimeter.
Following the example given:

• The center of the circle is \((2, -7)\).
• The point \((-2, -4)\) is on the circle.

By applying the distance formula:\[ r = \sqrt{(-2 - 2)^2 + (-4 + 7)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]This calculation gives us the radius \(r = 5\).
The process of computing the radius defines one of the circle's most important properties, influencing how the circle appears and behaves mathematically.