Problem 49 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, and ellipses. \(x^{2}+y^{2}-2 x+4 y-4=0\)

Step-by-Step Solution

Answer

The standard form is \((x-1)^2 + (y+2)^2 = 21\). The center is (1, -2) and the radius is \(\sqrt{21}\).

Step 1: Identify the Type of Equation

The given equation is of the form \(x^2 + y^2 + Dx + Ey + F = 0\). It represents a circle because both \(x\) and \(y\) are squared and have the same coefficient of 1.

Step 2: Complete the Square for x and y

To rewrite the equation in standard form, complete the square for both the \(x\) and \(y\) terms. For \(x\), take half of the coefficient of \(x\) (-2), square it to get 1, and add and subtract it. For \(y\), take half of the coefficient of \(y\) (4), square it to get 16, and add and subtract it.

Step 3: Reorganize the Equation

Incorporate the completed squares into the equation: \((x^2-2x+1) + (y^2+4y+16) -1 -16 -4 = 0 \) This simplifies to:\((x-1)^2 + (y+2)^2 = 21 \)

Step 4: Write the Equation in Standard Form

The standard form of a circle’s equation is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. The equation \((x-1)^2 + (y+2)^2 = 21\) indicates a center of \((1, -2)\) and a radius of \(\sqrt{21}\).

Step 5: Graph the Circle

Plot the center of the circle at \((1, -2)\) on a graph. Draw a circle with a radius \(\sqrt{21}\), which is approximately 4.6 units, around the center. Ensure the circle is symmetrical with respect to the center.

Key Concepts

Completing the Square

Completing the square is a technique used to simplify quadratic expressions. This method allows us to transform quadratic equations into a form that is easier to interpret, especially when dealing with circles. In our exercise, we need to rewrite the equation \(x^2 + y^2 - 2x + 4y - 4 = 0\) into a special format.

Here's how completing the square works:

• First, identify the terms involving \(x\) and \(y\). These terms need to be grouped separately. For \(x\), we look at \(x^2 - 2x\), and for \(y\), at \(y^2 + 4y\).
• To complete the square for the \(x\)-terms, take half of the coefficient of \(x\) (which is -2), and then square it. This gives \( (-1)^2 = 1\). Add and subtract this inside the expression: \( x^2 - 2x + 1 - 1\).
• Do the same for the \(y\)-terms by taking half of 4, squaring to get 16, and then adding and subtracting inside: \( y^2 + 4y + 16 - 16\).

By doing this for both \(x\) and \(y\), you can reorganize the equation into something much clearer: \((x-1)^2 + (y+2)^2 - 1 - 16 - 4 = 0\). The completed squares help us later turn the equation into its standard form.

Standard Form of a Circle

The standard form of a circle’s equation is crucial in geometry as it provides a straightforward way to find the circle's center and radius. This form is given by: \((x-h)^2 + (y-k)^2 = r^2\).

The elements of this equation are:

• \((h, k)\) representing the center of the circle. In our example problem, this center is \((1, -2)\), derived from the equation \((x-1)^2 + (y+2)^2 = 21\).
• \(r^2\), which represents the square of the radius. For our equation, \(r^2 = 21\), indicating that the radius \(r\) is \(\sqrt{21}\) or approximately 4.6.

When converting to this form, our goal is to clearly identify these components. The step-by-step solution achieved this by rearranging and completing the square to end up with a neat presentation. The standard form not only simplifies the graphing process but also enhances understanding of key geometric properties.

Graphing Circles

Graphing circles based on their equations can seem challenging at first, but it becomes straightforward once you know the standard form. Let’s use our equation \((x-1)^2 + (y+2)^2 = 21\) as an example.

To graph a circle:

• Start with plotting the center of the circle, which is at \((1, -2)\). This point is the anchor for your circle.
• Then, measure out the radius from this center point. The radius is \(\sqrt{21}\), approximately 4.6 units. Use a ruler or a compass if necessary.
• Draw the circle by ensuring it stays at this radius distance from the center all the way around. It should look perfectly round and symmetrical.

Remember, proper graphing involves accuracy, especially with distances and symmetry around the center. Once you draw it, you’ll see how the mathematical description aligns perfectly with the visual representation.