Problem 90 - Intermediate Algebra

Question

Explain the error in each statement. The graph of \(x^{2}+(y+3)^{2}=10\) is a circle with center \((0,-3)\) and radius 5

Step-by-Step Solution

Answer

The center is correct; the radius should be \(\sqrt{10}\), not 5.

Step 1: Identify the Equation Form

The given equation is \(x^2 + (y+3)^2 = 10\). This equation is in the form \((x-h)^2 + (y-k)^2 = r^2\), which represents the equation of a circle. Here, \(h=0\), \(k=-3\), and \(r^2=10\).

Step 2: Determine Center of Circle

In the standard form of the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the center is given by the point \((h, k)\). For \(x^2 + (y+3)^2 = 10\), the center is \((0, -3)\). This statement about the center is correct.

Step 3: Calculate Radius from Equation

The equation \((x-h)^2 + (y-k)^2 = r^2\) shows that \(r^2=10\), which means the radius \(r\) is calculated as \(r = \sqrt{10}\), not 5. The radius is the square root of the constant term. The statement about the radius is incorrect.

Key Concepts

Radius of a Circle

The radius of a circle is one of the fundamental components in the equation of a circle. It is the distance from the center of the circle to any point on its circumference. When you're dealing with the standard form of a circle equation, \[(x-h)^2 + (y-k)^2 = r^2,\] the value of the radius is derived from the constant term on the right-hand side of the equation, which is noted as \(r^2\).
To find the radius \(r\), you simply take the square root of \(r^2\). For example, if your equation looks like \(x^2 + (y+3)^2 = 10\), then \(r^2 = 10\), and thus the radius \(r = \sqrt{10}\).
Essential things to remember about the radius:

• The radius is always a positive number because it represents a distance.
• It determines the size of the circle: greater the radius, larger the circle.
• It should not be confused with \(r^2\), which is the number used in the equation.

Standard Form of a Circle

The standard form of a circle's equation is a neat and organized way to represent a circle's properties using algebra. It is written as: \[(x-h)^2 + (y-k)^2 = r^2,\] where:

• \(h\) and \(k\) identify the coordinates of the center of the circle.
• \(r\) represents the circle's radius.

This form is particularly useful because it clearly shows the central (\(h, k\)) and the circle's radius in a simple equation. You can quickly identify these properties just by looking at the equation without graphing the circle.
If you have an equation like \((x-3)^2 + (y+2)^2 = 25\), you can immediately see:

• The center is at \((3, -2)\).
• The radius \(r\) is \(\sqrt{25} = 5\).

Center of a Circle

The center of a circle serves as a crucial point from which all points on the circumference are equidistant, forming the circle. In the standard form of a circle's equation, \[(x-h)^2 + (y-k)^2 = r^2,\] the center is denoted by the point \((h, k)\).
To identify the center from an equation, simply locate the values (\(h\) and \(k\)) in the terms \((x-h)\) and \((y-k)\). If the equation reads \((x-1)^2 + (y+4)^2 = 36\), the center is at \((1, -4)\).

• The value of \(h\) is found by changing the sign of the number next to \(x\).
• Similarly, \(k\) is found by negating the number next to \(y\).

Key points about the center:

• It defines the position of the circle in the coordinate plane.
• It has no impact on the size of the circle, only the location.
• Understanding the center helps in graphing the circle precisely on a graph.