Problem 10 - A Graphical Approach to College Algebra

Question

Find the center-radius form for each circle satisfying the given conditions. Center \((-2,5) ;\) radius 4

Step-by-Step Solution

Answer

The equation is \((x + 2)^2 + (y - 5)^2 = 16\).

Step 1: Recall the Equation of a Circle

The standard equation of a circle in center-radius form is given by: \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify the Given Values

The center of the circle is given as \((-2, 5)\), so \(h = -2\) and \(k = 5\). The radius \(r\) is given as 4.

Step 3: Substitute Values into the Equation

Substitute \(h = -2\), \(k = 5\), and \(r = 4\) into the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\). This gives us: \((x + 2)^2 + (y - 5)^2 = 4^2\).

Step 4: Simplify the Radius Term

Calculate \(4^2\) to find the square of the radius: \(4^2 = 16\).

Step 5: Write the Final Equation

Replace \(4^2\) with 16 in the equation from Step 3 to get the final center-radius form: \((x + 2)^2 + (y - 5)^2 = 16\).

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is a fundamental concept in geometry that provides a simple way to represent all the points making up a circle on a coordinate plane. This equation is written as: \[ (x-h)^2 + (y-k)^2 = r^2 \] Here:

• \((h, k)\) represents the center of the circle. These are the coordinates where the circle is perfectly balanced, much like the center of a wheel.
• \(r\) is the radius of the circle. It's the constant distance from the center to any point on the circle's edge.

This form is also known as the center-radius form because it directly incorporates these key circle features. Understanding this simple equation allows you to easily graph circles by seeing their location and size at a glance.

Identify Circle Center and Radius

To use the standard equation of a circle, it's crucial to first identify the circle’s center and radius. In our exercise, the center of the circle was given as \((-2, 5)\). Here, the values \(h = -2\) and \(k = 5\) directly identify the circle's position.

• Center \((h, k)\): The coordinates \((-2, 5)\) tell us exactly where this circle is situated on the coordinate plane. The first number, \(-2\), is the \(x\)-coordinate, indicating it is to the left of the y-axis. The second number, \(5\), is the \(y\)-coordinate, showing it is above the x-axis.
• Radius \(r\): The radius is given as \(4\). This tells us how far the circle stretches from its center in any direction.

Correctly identifying these values is the first step in transitioning to the center-radius form of the circle equation.

Substitute Values into Equation

After identifying the center \((-2, 5)\) and the radius \(4\), the next step is to substitute these values into our standard circle equation \[(x-h)^2 + (y-k)^2 = r^2\]. Here's how it looks:

• Replace \(h\) with \(-2\). Since subtracting a negative is equivalent to adding, the term becomes \((x + 2)^2\).
• Replace \(k\) with \(5\). This gives us \((y - 5)^2\).
• Replace \(r\) with \(4\). So \(r^2\) becomes \(4^2\), simplifying further to \(16\).

Thus, the resulting equation in center-radius form is \[(x + 2)^2 + (y - 5)^2 = 16\]. This equation represents all the points \((x, y)\) that form a circle with a center at \((-2, 5)\) and a radius of \(4\). It concisely captures the geometry of a circle in a neat mathematical form.