Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((-1,2) ;\) passing through \((2,6)\)

Step-by-Step Solution

Answer

The circle's equation is \((x + 1)^2 + (y - 2)^2 = 25\).

Step 1: Recall the center-radius form of a circle

The equation for the center-radius form 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.

Step 2: Identify the known values

From the problem, the center of the circle \((h, k)\) is \((-1, 2)\). The point that lies on the circle is \((2, 6)\).

Step 3: Calculate the radius

The radius \(r\) of the circle is the distance between the center \((-1, 2)\) and the point \((2, 6)\). Calculate this distance using the distance formula \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where \((x_1, y_1)\) is the center and \((x_2, y_2)\) is the point on the circle.

Step 4: Apply the distance formula

Substitute \((x_1, y_1) = (-1, 2)\) and \((x_2, y_2) = (2, 6)\) into the distance formula: \[ r = \sqrt{(2 - (-1))^2 + (6 - 2)^2} \] \[ r = \sqrt{(2 + 1)^2 + (4)^2} \] \[ r = \sqrt{3^2 + 4^2} \] \[ r = \sqrt{9 + 16} \] \[ r = \sqrt{25} \] \[ r = 5 \].

Step 5: Write the center-radius form of the equation

With the center \((-1, 2)\) and the radius \(r = 5\), substitute these into the center-radius form: \[ (x - (-1))^2 + (y - 2)^2 = 5^2 \] \[ (x + 1)^2 + (y - 2)^2 = 25 \].

Key Concepts

Center-Radius Form

The center-radius form is a simple way to express the equation of a circle, maximizing understanding of the circle's basic components: its center and its radius. This form is presented as \( (x - h)^2 + (y - k)^2 = r^2 \), where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

To apply this, identify the center point of the circle, \((h, k)\), and calculate the distance from the center to any point on the circle to find \(r\). This approach helps visualize the circle’s position and size in the coordinate plane, anchoring its equation in geometric properties.

Distance Formula

Understanding how to calculate the distance between two points in a coordinate plane is crucial for finding a circle's radius. The distance formula is derived from the Pythagorean theorem and is written as:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula helps measure the straight line distance between two points \((x_1, y_1)\) and \((x_2, y_2)\). For circles, this is helpful in determining the radius when a point on the circle's circumference and the center are known. Simply plug the coordinates into the formula, execute the operations, and you'll find the radius or any required distance on the plane. This calculation is foundational when constructing the center-radius form of a circle equation.

Radius Calculation

The radius is a fundamental aspect of a circle, dictating its size and shape. Calculating the radius involves finding the distance between the circle's center and any point on its circumference. Given the center at \((-1, 2)\) and a point \((2, 6)\) that lies on the circle, you use the distance formula:\[r = \sqrt{(2 - (-1))^2 + (6 - 2)^2}\]Step through:

• Calculate \((2 + 1)^2 = 3^2 = 9\).
• Calculate \((4)^2 = 16\).
• Add them: \(9 + 16 = 25\).
• Find the square root: \(\sqrt{25} = 5\).

So, the radius \(r\) is \(5\). This precise calculation leads to the complete circle equation, serving as a practical example of applying mathematical theories to find exact measurements in geometry.