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### Course: Digital SAT Math>Unit 13

Lesson 6: Circle equations: advanced

# 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 $xy$-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 $\left(x+2{\right)}^{2}+\left(y-3{\right)}^{2}={4}^{2}$ is graphed in the $xy$-plane below. It is a circle with a center at $\left(-2,3\right)$ 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

Khan Academy video wrapper
Features of a circle from its standard equationSee video transcript

### Representing a circle in the $xy$‍ -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 $xy$-plane, a circle with center $\left(h,k\right)$ and radius $r$ has the equation:
$\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$
For example, the circle above has a center located at $\left(1,2\right)$ and a radius of $3$. For $h=1$, $k=2$, and $r=3$, its equation is:
$\left(x-1{\right)}^{2}+\left(y-2{\right)}^{2}={3}^{2}$

### Try it!

try: find a circle that meets a criterion
If all points on a circle
in the $xy$-plane, which of the following could be the equation of the circle?
Choose 1 answer:

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

### How do I complete the square?

Khan Academy video wrapper
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 ${x}^{2}+2x+{y}^{2}-10y+22=0$ to guide us through how to rewrite an expanded equation in standard form.
First, we have to make sure the coefficients of ${x}^{2}$ and ${y}^{2}$ 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 ${x}^{2}+2x$, 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 $\frac{1}{2}$ 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 $\frac{1}{2}$ the coefficient.
The coefficient of the $x$-term is $2$. Therefore, the constant ${\left(\frac{2}{2}\right)}^{2}=1$ completes the square for $x$:
${x}^{2}+2x+1=\left(x+1{\right)}^{2}$
The coefficient of the $y$-term is $-10$. Therefore, the constant ${\left(\frac{-10}{2}\right)}^{2}=25$ completes the square for $y$:
${y}^{2}-10y+25=\left(y-5{\right)}^{2}$
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.
$\begin{array}{rl}{x}^{2}+2x+{y}^{2}-10y+22& =0\\ \\ {x}^{2}+2x+1+{y}^{2}-10y+25+22& =0+1+25\\ \\ \left(x+1{\right)}^{2}+\left(y-5{\right)}^{2}+22& =26\end{array}$
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.
$\begin{array}{rl}\left(x+1{\right)}^{2}+\left(y-5{\right)}^{2}+22& =26\\ \\ \left(x+1{\right)}^{2}+\left(y-5{\right)}^{2}+22-22& =26-22\\ \\ \left(x+1{\right)}^{2}+\left(y-5{\right)}^{2}& =4\\ \\ \left(x+1{\right)}^{2}+\left(y-5{\right)}^{2}& ={2}^{2}\end{array}$
The equation represents a circle with a center at $\left(-1,5\right)$ 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 ${x}^{2}$-term and the ${y}^{2}$-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
${x}^{2}-12x+{y}^{2}+2y=13$
The equation above represents a circle in the $xy$-plane.
What number, when added to ${x}^{2}-12x$, gives us an expression that can be factored into $\left(x-6{\right)}^{2}$ ?
What number, when added to ${y}^{2}+2y$, gives us an expression that can be factored into $\left(y+1{\right)}^{2}$ ?

## Your turn!

practice: identify the equation of a circle in standard form
A circle in the $xy$-plane has center $\left(-3,-4\right)$, and radius $5$. Which of the following is an equation of the circle?
Choose 1 answer:

Practice: identify a circle's diameter from equation
$\left(x-3{\right)}^{2}+\left(y-3{\right)}^{2}=6$
The equation above defines a circle in the $xy$-plane. What is the diameter of the circle?
Choose 1 answer:

Practice: interpret a circle equation not in standard form
${x}^{2}+6x+{y}^{2}-4y=3$
The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?
Choose 1 answer:

## Things to remember

In the $xy$-plane, a circle with center $\left(h,k\right)$ and radius $r$ has the equation:
$\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$

## Want to join the conversation?

• hey this is from march 2023, good luck loves!
(160 votes)
• hey this is from october 5 2023, thank u!!
(54 votes)
• you got this! you are ready for the sat and you will ace it!
(87 votes)
• amen in Jesus name thnks brr
(11 votes)
• idk i find it cute when people around the world who's taking tomorrow's sat are panicking cooperatively rn. good luck bro
(118 votes)
• ahhhh finally something easy
(40 votes)
• bruh wha
(5 votes)
• August 26 ?
(32 votes)
• YES.. here we go. and yes calculators are allowed for both sections
(10 votes)
• Who all are giving SAT on October 7th?
(28 votes)
• me me meee
(4 votes)
• Anybody taking SAT tom on Aug 26? Good luck yall. Don't panic! (I'm panicking tho.Lol)
(22 votes)
• math is easy
(20 votes)
• true. you just need extreme focus. no distractions, but most importantly good memory although I think everyone has good memory, they just need to stop doing certain things haha
(2 votes)
• Tips: when you are determining if the equation of circle lies in the positive quadrant or not

check h, k and r
h,k>r for circle to be in +ve quad
(19 votes)
• good luck to anyone giving it on Aug 24, 2024!!
(17 votes)