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## Digital SAT Math

### Course: Digital SAT Math>Unit 5

Lesson 6: Circle equations: foundations

# Circle equations | Lesson

A guide to circle equations on the digital SAT

## What are circle equations, and how frequently do they appear on the test?

We can describe circles in the x, y-plane using equations in terms of x and y. Circle equations questions require us to understand the connection between these equations and the features of circles.
For example, the equation left parenthesis, x, plus, 2, right parenthesis, squared, plus, left parenthesis, y, minus, 3, right parenthesis, squared, equals, 4, squared is graphed in the x, y-plane below. It is a circle with a center at left parenthesis, minus, 2, comma, 3, right parenthesis and a radius of 4.
A circle with the equation (x+2)^2+(y-3)^2=4^2 is graphed in the xy-plane. The circle has a center located at (-2, 3) and a radius of 4.
In this lesson, we'll learn to:
1. Relate the standard form equation of a circle to the circle's center and radius
2. Rewrite a circle equation in standard form by completing the square
Note: To solve the harder circle equations problems, we need to know how rewrite quadratic expressions by completing the square.
You can learn anything. Let's do this!

## What is the standard form equation of a circle?

### Features of a circle from its standard equation

Features of a circle from its standard equationSee video transcript

### Representing a circle in the $xy$x, y-plane

Just like the equations for lines and parabolas, the standard form equation of a circle tells us about the circle's features.
In the x, y-plane, a circle with center left parenthesis, h, comma, k, right parenthesis and radius r has the equation:
left parenthesis, x, minus, h, right parenthesis, squared, plus, left parenthesis, y, minus, k, right parenthesis, squared, equals, r, squared
A circle in the xy-plane has a center located at (1, 2) and a radius extending from (1, 2) to (4, 2). The center is labeled (h,k), and the radius is r units long.
For example, the circle above has a center located at left parenthesis, 1, comma, 2, right parenthesis and a radius of 3. For start color #7854ab, h, equals, 1, end color #7854ab, start color #ca337c, k, equals, 2, end color #ca337c, and start color #208170, r, equals, 3, end color #208170, its equation is:
left parenthesis, x, minus, start color #7854ab, 1, end color #7854ab, right parenthesis, squared, plus, left parenthesis, y, minus, start color #ca337c, 2, end color #ca337c, right parenthesis, squared, equals, start color #208170, 3, end color #208170, squared

### Try it!

try: find a circle that meets a criterion
If all points on a circle
in the x, y-plane, which of the following could be the equation of the circle?

## How do I rewrite equations of circles in standard form?

### How do I complete the square?

Features of a circle from its expanded equationSee video transcript

### Rewriting circle equations in standard form by completing the square

The standard form equation of a circle contains the squares of two binomials. Sometimes, we'll be asked to determine the center or radius of a circle represented by an equation in which the squares of the binomials are expanded.
Let's use the expanded equation x, squared, plus, 2, x, plus, y, squared, minus, 10, y, plus, 22, equals, 0 to guide us through how to rewrite an expanded equation in standard form.
First, we have to make sure the coefficients of x, squared and y, squared are both 1. For most questions that require completing the square on the SAT, the coefficients will be 1.
Next, we need to find the constants that complete the square for x and y. For x, this means we need to find a constant that, when added to x, squared, plus, 2, x, lets us rewrite the expression as the square of a binomial.
You'll naturally develop a sense for constants that complete the square as you work on polynomial multiplication and factoring. A shortcut is to remember that the constant term of the binomial is equal to start fraction, 1, divided by, 2, end fraction the coefficient of the x- or y-term, and the constant that needs to be added to complete the square is equal to the square of start fraction, 1, divided by, 2, end fraction the coefficient.
The coefficient of the x-term is 2. Therefore, the constant left parenthesis, start fraction, 2, divided by, 2, end fraction, right parenthesis, squared, equals, 1 completes the square for x:
x, squared, plus, 2, x, plus, 1, equals, left parenthesis, x, plus, 1, right parenthesis, squared
The coefficient of the y-term is minus, 10. Therefore, the constant left parenthesis, start fraction, minus, 10, divided by, 2, end fraction, right parenthesis, squared, equals, 25 completes the square for y:
y, squared, minus, 10, y, plus, 25, equals, left parenthesis, y, minus, 5, right parenthesis, squared
We can rewrite the equation as shown below. Remember that when we add constants to one side of the equation, we must also add the same constants to the other side of the equation to keep the two sides equal.
\begin{aligned} x^2+2x+y^2-10y+22&=0 \\\\ x^2+2x\purpleD{+1}+y^2-10y\maroonD{+25}+22 &=0\purpleD{+1}\maroonD{+25} \\\\ (x+1)^2+(y-5)^2+22 &=26 \end{aligned}
Now that we have our completed squares, we just need to subtract 22 from both sides of the equation. The resulting constant on the right side of the equation is equal to the square of the radius.
\begin{aligned} (x+1)^2+(y-5)^2+22 &=26 \\\\ (x+1)^2+(y-5)^2+22\purpleD{-22} &=26 \purpleD{-22} \\\\ (x+1)^2+(y-5)^2 &=4 \\\\ (x+1)^2+(y-5)^2 &=2^2 \end{aligned}
The equation represents a circle with a center at left parenthesis, minus, 1, comma, 5, right parenthesis and a radius of 2.
To rewrite an expanded circle equation in standard form:
1. If necessary, divide both sides of the equation by the same number so that the coefficients of both the x, squared-term and the y, squared-term are 1.
2. Find the constant the completes the square for x.
3. Repeat step 2 for y.
4. Add the constants from steps 2 and 3 to both sides of the equation.
5. Rewrite the expanded expressions as the squares of binomials.
6. Combine the remaining constants on the right side of the equation. It is equal to the square of the radius.

### Try it!

try: complete the square in an expanded circle equation
x, squared, minus, 12, x, plus, y, squared, plus, 2, y, equals, 13
The equation above represents a circle in the x, y-plane.
What number, when added to x, squared, minus, 12, x, gives us an expression that can be factored into left parenthesis, x, minus, 6, right parenthesis, squared ?
What number, when added to y, squared, plus, 2, y, gives us an expression that can be factored into left parenthesis, y, plus, 1, right parenthesis, squared ?

practice: identify the equation of a circle in standard form
A circle in the x, y-plane has center left parenthesis, minus, 3, comma, minus, 4, right parenthesis, and radius 5. Which of the following is an equation of the circle?

Practice: identify a circle's diameter from equation
left parenthesis, x, minus, 3, right parenthesis, squared, plus, left parenthesis, y, minus, 3, right parenthesis, squared, equals, 6
The equation above defines a circle in the x, y-plane. What is the diameter of the circle?

Practice: interpret a circle equation not in standard form
x, squared, plus, 6, x, plus, y, squared, minus, 4, y, equals, 3
The equation above defines a circle in the x, y-plane. What are the coordinates of the center of the circle?