Problem 16 - A Graphical Approach to College Algebra

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 equation of the circle is \((x - 2)^2 + (y + 7)^2 = 25\).

Step 1: Identify Known Values

We are given the center of the circle as \((2, -7)\) and a point on the circle as \((-2, -4)\). The center-radius form of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Calculate the Radius

To find the radius, we calculate the distance between the center \((2, -7)\) and the point that lies on the circle \((-2, -4)\). The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting in our values: \[ r = \sqrt{(-2 - 2)^2 + (-4 + 7)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5\]So, the radius \(r\) is 5.

Step 3: Write the Center-Radius Form Equation

Now that we have the radius, we can write the equation of the circle in center-radius form. Substituting \(h = 2\), \(k = -7\), and \(r = 5\) into the formula \((x-h)^2 + (y-k)^2 = r^2\), we get: \[(x - 2)^2 + (y + 7)^2 = 5^2\]Which simplifies to:\[(x - 2)^2 + (y + 7)^2 = 25\]This is the center-radius form of the circle.

Key Concepts

Circle Equation

A circle equation is a vital concept in geometry that helps us describe a circle in a coordinate plane. The most common form of a circle equation is the center-radius form, which is written as \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the coordinates of the center of the circle, and \(r\) represents the radius, the distance from the center to any point on the circle.

To fully understand the circle equation, it helps to visualize each part of it:

• \( (x - h) \) and \( (y - k) \) are expressions that reflect how far a general point \((x, y)\) on the circle is horizontally and vertically displaced from the center \((h, k)\).
• The equation \((x - h)^2 + (y - k)^2\) represents the square of the distance from the center to any point on the circle.
• \(r^2\) is simply the square of the radius, which means the radius times itself.

To find the circle equation, you need to know the circle's center and its radius. Once these are known, you can quickly plug them into the center-radius form and solve for the equation.

Distance Formula

The distance formula is a straightforward algebraic tool used to calculate the distance between two points in a coordinate plane. It is particularly useful when finding the radius of a circle, as seen in this exercise.

The formula is:\[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

In the context of this problem, \((x_1, y_1)\) represents the center of the circle \((2, -7)\), and \((x_2, y_2)\) is a point on the circle \((-2, -4)\). The distance formula lets us calculate the length of the line segment that connects these two points, which is also the radius of the circle.

This formula arises from the Pythagorean theorem, which states that in a right-angled triangle:

• the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this scenario, the differences \((x_2 - x_1)\) and \((y_2 - y_1)\) represent the length of the horizontal and vertical legs of a right triangle, respectively. By squaring these differences, summing them, and taking the square root, we obtain the distance, or radius, \(r\). For example, using our points from the exercise, the distance formula calculates the radius as 5. This step is essential to establishing the circle equation.

Radius Calculation

Radius calculation is a critical step in writing the equation of a circle. It is the measure of how far the circle extends from its center to any point on its edge. Understanding and calculating the radius accurately allows us to describe the size of the circle mathematically.

In this exercise, we've already determined the radius using the distance formula. The given center of the circle is \((2, -7)\), and the point on the circle is \((-2, -4)\). Therefore, the radius is calculated as:
\[\sqrt{(-2 - 2)^2 + (-4 + 7)^2} = \sqrt{25} = 5\]To perform this calculation, you:

• Subtract the x-coordinate of the center from the x-coordinate of a point on the circle \((-2 - 2)\).
• Subtract the y-coordinate of the center from the y-coordinate of a point on the circle \((-4 + 7)\).
• Square these differences, add them together, and take the square root of the sum to give the radius \(r\).

Understanding the radius is important, as it feeds directly into the circle equation \((x - h)^2 + (y - k)^2 = r^2\). With the radius determined as 5, it becomes possible to write the complete equation of the circle.