Problem 31 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$x^{2}+y^{2}+2 x-4 y=4$$

Step-by-Step Solution

Answer

The circle's center is \((-1, 2)\) and its radius is 3.

Step 1: Write Down the Equation

The given equation is already provided: \[x^2 + y^2 + 2x - 4y = 4\] Start with this equation for simplification and finding the circle's properties.

Step 2: Rearrange Terms

Rearrange the equation by grouping x terms and y terms together:\[x^2 + 2x + y^2 - 4y = 4\]

Step 3: Complete the Square for x

To complete the square for the x terms, take half of the coefficient of x (which is 2), square it, and add it to both sides:\[x^2 + 2x \to (x + 1)^2 - 1\]Thus, rewrite it as:\[ (x + 1)^2 \]

Step 4: Complete the Square for y

Similarly, complete the square for the y terms. Take half of the coefficient of y (-4), square it, and add it to both sides:\[y^2 - 4y \to (y - 2)^2 - 4\]Thus, rewrite it as:\[ (y - 2)^2 \]

Step 5: Simplify the Equation

Substitute the completed squares into the equation and move constants to the other side:\[(x + 1)^2 - 1 + (y - 2)^2 - 4 = 4\]Simplifying, you get:\[ (x+1)^2 + (y-2)^2 = 9 \]

Step 6: Identify the Center and Radius

The standard equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.From the equation \[(x+1)^2 + (y-2)^2 = 9\], we identify:Center: \((h, k) = (-1, 2)\)Radius: \(r = \sqrt{9} = 3\).

Step 7: Graph the Circle

With the center \((-1, 2)\) and radius \(3\), plot the circle on a graph. Mark the center, then draw the circle ensuring it extends 3 units in all directions from the center.

Key Concepts

Completing the Square

Completing the square is a handy technique used in algebra to simplify quadratic expressions. It can turn a quadratic equation into a form that is easily recognizable, such as the circle equation form.

To complete the square for a variable like x, you follow these steps:

• Take the linear coefficient (the number in front of x), divide it by 2, and then square the result. For example, with the equation term 2x, compute \(\left(\frac{2}{2}\right)^2 = 1\).
• Add and subtract this squared number inside the equation to balance it out.
  This allows you to rewrite the quadratic as a binomial square.

In the given example, we had \(x^2 + 2x\), which was transformed to \( (x + 1)^2 - 1\). Similarly, do this for the y terms, which resulted in \( (y - 2)^2 - 4\). Completing the square makes solving and graphing the equation easier by converting it into a standard recognizable form.

Center of a Circle

The center of a circle in a graph is identified using the standard form of the circle equation: \( (x-h)^2 + (y-k)^2 = r^2\). Here, \(h\) and \(k\) are the coordinates of the center of the circle.

When an equation is rearranged to match this form, it's easy to pull out the center's coordinates. For instance, in the equation \( (x + 1)^2 + (y - 2)^2 = 9\), the center is at \( (-1, 2)\).

This is found by understanding that the expression \( (x+1)\) indicates a \(h\) value of \(-1\), as we define \(x + 1\) to mean \(x - (-1)\). Similarly, \( (y - 2)\) shows that \(k\) equals \2\, confirming the center at \((-1, 2)\).
This concept is important for graphing, as the center is the point from which the radius extends.

Radius of a Circle

The radius is the distance from the center of a circle to any point on its circumference. In mathematical terms, it's denoted by \(r\) in the equation \( (x-h)^2 + (y-k)^2 = r^2\). Finding the radius involves taking the square root of the number on the right side of the circle's equation.

From the equation \( (x+1)^2 + (y-2)^2 = 9\), the radius squared is \9\. Therefore, the radius, \(r\), equals \(\sqrt{9}\), which is \3\. This tells us how far the circle extends from the center in all directions.

Understanding the radius not only helps us predict the size of the circle but is also essential in graphing it correctly on a coordinate plane. The circle's size will depend on the length of the radius, which is a direct measure of its geometric boundaries.

Graphing Circles

Graphing a circle starts with identifying its center and radius from the circle equation. With these two pieces of information, you can accurately plot the circle on a coordinate plane.

Here's how to graph the circle once you have the equation in standard form:

• Locate the center of the circle on the graph using the \(h\) and \(k\) values from the equation \( (x-h)^2 + (y-k)^2 = r^2\). In the example \( (x+1)^2 + (y-2)^2 = 9\), the center is \((-1, 2)\).
• Draw a point on the graph at this location as your starting point.
• From the center, measure outwards by the value of the radius (in our case, 3 units), drawing a circle that extends this distance in all directions. This creates the circle's boundary on the graph.

Visualizing a circle graphically helps understand its spatial aspects, like where it lies on the plane and its relative size compared to other figures. Connecting these concepts allows for more complex spatial reasoning in geometry.