Problem 33 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$(x+2)^{2}+(y-3)^{2}=7$$

Step-by-Step Solution

Answer

The center of the circle is (-2, 3) and the radius is \(\sqrt{7}\).

Step 1: Identify the Equation of a Circle

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

Step 2: Determine the Center of the Circle

Compare the given equation \((x+2)^2 + (y-3)^2 = 7\) with \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h = -2\) and \(k = 3\). Therefore, the center of the circle is \((-2, 3)\).

Step 3: Calculate the Radius

The right side of the equation \((x+2)^2 + (y-3)^2 = 7\) is equal to \(r^2\). Therefore, \(r^2 = 7\). Taking the square root of both sides gives \(r = \sqrt{7}\).

Step 4: Graph the Circle

To graph the circle, start at the center \((-2, 3)\). Use the radius \(\sqrt{7}\) (approximately 2.65) to mark points at that distance from the center in all directions. Draw a smooth curve connecting these points to represent the circle.

Key Concepts

Center of a Circle

In the equation of a circle \((x-h)^2 + (y-k)^2 = r^2\), the parts \(h\) and \(k\) play crucial roles in determining the center of the circle. Previously, the general formula tells us that the center of the circle is located at point \((h, k)\). This means:

• If \(h\) is a negative number, it shifts the center along the x-axis to the left.
• If \(k\) is a positive number, this shifts the center along the y-axis upward.

Let's use the provided equation: \((x+2)^2 + (y-3)^2 = 7\). Here, the center is not immediately obvious, but by transforming \(x + 2\) into \(x - (-2)\), we determine \(h = -2\) and \(k = 3\). Thus, we find that the circle's center is at \((-2, 3)\). This is important for understanding where to place the circle on a graph. The center is your starting point and will guide all other placements for a correct graphical representation.

Radius of a Circle

The radius is a key component of the circle's equation. It affects the size and reach of the circle in the coordinate plane. According to the standard form \((x-h)^2 + (y-k)^2 = r^2\), the term \(r\) reflects the radius. However, in the given equation, \((x+2)^2 + (y-3)^2 = 7\), we compare \(r^2\) directly against 7.To find the radius, simply take the square root of the number on the right side of the equation. Therefore, \(r = \sqrt{7}\), which is approximately 2.65 when rounded to two decimal places. It may not be a whole number, but this approximate value is still crucial. The radius signifies the distance from the center of the circle to any point on its circumference.

Graphing Circles

Graphing a circle involves plotting it accurately on the coordinate plane, and it’s more understandable once the center and radius are known. Start with the circle equation \((x+2)^2 + (y-3)^2 = 7\). First, plot the center at the point \((-2, 3)\). This acts as your anchor point. Once the center is marked, use the radius. With \(r = \sqrt{7}\), which is roughly 2.65, you can measure this distance in all directions from the center.Imagine moving from the center:

• Two and a half units to the right and left along the x-axis
• Two and a half units up and down along the y-axis

Mark these points as a reference. From there, sketch a smooth curve that connects these points into a circle. This visual step ensures you accurately represent how far the circle extends from the center in all directions.