Problem 90 - College Algebra and Calculus: An Applied Approach

Question

Find the standard form of the equation of the specified circle. Center: \((3,-2) ;\) point on circle: \((-1,1)\)

Step-by-Step Solution

Answer

So, the standard form of the equation of the circle that has center at \((3,-2)\) and passes through the point \((-1,1)\) is \((x-3)^2 + (y+2)^2 = 25\).

Step 1: Find the radius using the distance formula

The distance between two points \((x1, y1)\) and \((x2, y2)\) can be calculated using the formula \(\sqrt{(x2-x1)^2+(y2-y1)^2}\). Here the two points are the center of the circle \((3,-2)\) and a point on the circle \((-1,1)\). Applying these to the formula gives \(\sqrt{(3-(-1))^2+(-2-1)^2} = \sqrt{(4)^2+(-3)^2} = \sqrt{16+9} = \sqrt{25} = 5\). So, the radius of the circle is 5.

Step 2: Substitute the values into the standard form

Substitute \(h=3\), \(k=-2\), and \(r=5\) into the standard form equation \((x-h)^2 + (y-k)^2 = r^2\). This gives \((x-3)^2 + (y+2)^2 = 5^2\).

Key Concepts

Equation of a Circle

The equation of a circle in standard form is a way to express all the possible points that make up the circle in the coordinate plane. The standard form of the equation is \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) represent the coordinates of the center of the circle, while \(r\) stands for the radius.
This equation is derived from the distance formula but adapted specifically for circles:

• \((x-h)\) refers to the horizontal distance from the center of the circle.
• \((y-k)\) refers to the vertical distance from the center.
• \(r^2\) is the square of the radius, representing how far all points are from the center.

This form not only defines the circle but immediately provides insights into its geometrical properties.

Distance Formula

The distance formula is crucial for finding the radius of a circle when given two points: the center and any point on the circumference. It is also derived from the Pythagorean theorem and is written as \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).
Here's how you can use it:

• Identify your given points: \(x_1, y_1\) and \(x_2, y_2\).
• Subtract the respective coordinates: calculate \(x_2-x_1\) and \(y_2-y_1\).
• Square the results to eliminate negative values.
• Add them together and take the square root of the sum to find the distance.

In the context of a circle, this distance is the radius, which measures how far all points on the boundary are from the center.

Radius Calculation

To find the radius of the circle, you need a specific point on the circle and the center of the circle itself. The radius is essentially the distance from the center to the given point.
Applying the distance formula:

• Use the coordinates of the center, which are \(h = 3, k = -2\).
• Pair them with the point on the circle, \((-1, 1)\).
• After substitution into the distance formula \(\sqrt{(3-(-1))^2+(-2-1)^2}\), you perform the calculations \(\sqrt{4^2 + (-3)^2}\).
• This results in \(\sqrt{16 + 9} = \sqrt{25} = 5\), establishing the radius as 5 units.

This process allows you to plug the exact radius back into the circle's equation for precise representation.

Center of a Circle

The center of a circle is a pivotal part of understanding both the equation and the geometry of the circle itself. Denoted by the coordinates \(h\) and \(k\), the center is the point where all radial lines (radii) converge.
The center holds a critical function:

• It is the reference point for determining every position on the circle. All points on the circle are equidistant from this central point.
• In a circle's equation, the center's coordinates replace \(h\) and \(k\), organizing the spatial orientation of the circle in the plane.

In our example, the center is located at \(3, -2\), and it directly influences the circle's placement within the coordinate grid.