Problem 35 - Intermediate Algebra

Question

Write each equation of a circle in standard form and graph it. Give the coordinates of its center and give the radius. $$x^{2}+y^{2}-2 x+4 y=-1$$

Step-by-Step Solution

Answer

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

Step 1: Rearrange Equation

Start by isolating the x and y terms on one side of the equation: \[ x^2 - 2x + y^2 + 4y = -1 \]

Step 2: Complete the Square for x

Complete the square for the x terms. Take the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\).Add and subtract this square inside the equation:\[ (x^2 - 2x + 1) + y^2 + 4y = -1 + 1 \] This simplifies to:\[ (x-1)^2 + y^2 + 4y = 0 \]

Step 3: Complete the Square for y

Complete the square for the y terms. Take the coefficient of \(y\), which is \(4\), divide it by 2 to get \(2\), and then square it to get \(4\).Add and subtract this square in the equation:\[ (x-1)^2 + (y^2 + 4y + 4) = 0 + 4 \]This simplifies to:\[ (x-1)^2 + (y+2)^2 = 4 \].

Step 4: Identify the Circle's Center and Radius

The equation \((x-1)^2 + (y+2)^2 = 4\) is in standard form, \((x-h)^2 + (y-k)^2 = r^2\), where \( (h, k) \) is the center, and \( r \) is the radius of the circle. Thus, the center is \( (1, -2) \), and the radius is \( r = \sqrt{4} = 2 \).

Key Concepts

Standard Form

The standard form of a circle's equation is an organized method for expressing the circle's equation that reveals important features, like its center and radius, at a glance. This is represented as \((x-h)^2 + (y-k)^2 = r^2\), where

• \(h\) and \(k\) are the x and y coordinates of the circle's center, respectively.
• \(r\) is the radius of the circle.

To convert an equation into this form, such as \(x^2 + y^2 - 2x + 4y = -1\), you'll arrange the terms into groupings that allow for completing the square. Once in this form, identifying the circle's characteristics becomes much easier.

Completing the Square

Completing the square is an essential mathematical process used to create perfect square trinomials, thereby rewriting quadratic expressions in a squared binomial form. This is crucial for converting a circle's equation into its standard form.
To complete the square for

• the \(x\) terms, take the coefficient of \(x\), which is -2. Divide it by 2 to get -1. Then square it to find 1, which you add and subtract to the equation:
• the \(y\) terms, take the coefficient of \(y\), which is 4. Divide by 2 to get 2, then square it to get 4. Add and subtract this to transform the expression.

This method allows us to transform terms into \((x-1)^2 + (y+2)^2\), facilitating the recognition of the circle's center and radius.

Coordinates of Center

The center of a circle is a fixed point equidistant from all points on the perimeter of the circle. Recognizing this point is fundamental when working with circles. Once the equation is in standard form, \((x-h)^2 + (y-k)^2 = r^2\), you can directly read off the coordinates of the center as \((h, k)\).
In the final form of the equation, \((x-1)^2 + (y+2)^2 = 4\), the coordinates of the center are given as \((1, -2)\).
Understanding how to derive the center quickly allows for efficient graphing and a clearer comprehension of the circle's location on the Cartesian plane.

Graphing Circles

Graphing a circle involves plotting the circle on a coordinate grid, using its center and radius. This visual representation not only aids in understanding spatial relationships but also enhances problem-solving skills.
To graph a circle like \((x-1)^2 + (y+2)^2 = 4\):

• First, identify the center \((1, -2)\).
• Draw this point on the graph.
• Next, recognize the radius as \(r = 2\).
• Then, measure 2 units in all directions (up, down, left, right) from the center to determine points that lie on the circle.
• Finally, connect these points to sketch the circle.

This process results in an accurate graphical representation, helping in visualizing the circle's dimensions and placement on the plane.