Problem 25 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ (x-3)^{2}+(y+7)^{2}=50 $$

Step-by-Step Solution

Answer

Center: (3, -7); Radius: \(5\sqrt{2}\).

Step 1: Identify the Circle Equation Form

The given equation \((x-3)^2 + (y+7)^2 = 50\) is in the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\). In this form, \((h, k)\) is the center of the circle, and \(r\) is the radius.

Step 2: Determine the Center of the Circle

From the equation \((x-3)^2 + (y+7)^2 = 50\), compare with the standard form to extract the center. Here, \(h = 3\) and \(k = -7\), so the center of the circle is \((3, -7)\).

Step 3: Calculate the Radius

The equation \((x-3)^2 + (y+7)^2 = 50\) shows that \(r^2 = 50\). To find the radius \(r\), take the square root of 50: \(r = \sqrt{50} = 5\sqrt{2}\).

Step 4: Graph the Circle

To graph the circle, start by plotting the center at \((3, -7)\). Then, using the radius \(5\sqrt{2}\), draw a circle around this center point. This radius corresponds to approximately 7.07 units, so ensure the circle extends to that distance from the center in all directions.

Key Concepts

Center of Circle

When working with circle equations, identifying the center of the circle is a crucial first step. Let's explore why this is important and how it is calculated.
In a standard circle equation, such as \[(x-h)^2 + (y-k)^2 = r^2\]we observe that the center of the circle is represented by the coordinates \((h, k)\). This means the values of \(h\) and \(k\) are taken from the translated values of \(x\) and \(y\) in the equation.
In the given equation \((x-3)^2 + (y+7)^2 = 50\), we can identify the center by comparing it to the standard form. Here, the transformation involves subtracting 3 from \(x\) and adding 7 to \(y\). Therefore, the center is at the point \((3, -7)\).
Why is this important? Having the center allows us to know exactly where the circle is located on a graph, which is fundamental when graphing or analyzing intersections with other shapes.

Radius of Circle

Understanding the radius of a circle is another key concept when dealing with circle equations.
The radius is essentially the distance from the center of the circle to any point on its boundary. In a standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), the radius is the square root of the value on the right side of the equation.
In the given example \((x-3)^2 + (y+7)^2 = 50\), the equation shows \(r^2 = 50\), making the radius \(r = \sqrt{50}\). Simplifying this, we get\(r = 5\sqrt{2}\), approximately equal to 7.07. This means each point on the circle is 7.07 units away from the center at \((3, -7)\).
Having knowledge of the radius allows us to correctly draw the circle and understand its size, which is vital in applications like area calculations and determining points of intersections with other geometrical shapes.

Graphing Circles

Graphing a circle on a coordinate plane might seem challenging at first, but it becomes straightforward with knowledge of the center and radius.
Here’s a quick guide to graphing circles:

• Start by plotting the center of the circle on the graph. In this example, place a point at \((3, -7)\).
• Using the radius, \(5\sqrt{2}\) (approximately 7.07), measure outward from the center in all directions to mark the boundary of the circle.
• Draw the curve that connects these points to form a circle. Ensure that every point on the line is exactly 7.07 units away from the center.

Graphing is a useful skill for visualizing equations and understanding mathematical relationships. A well-drawn graph can illustrate the circle’s position and size, providing insights that are difficult to discern from the equation alone.