In the -plane, the circle is defined by the equation Which of the following gives the center of the circle and its radius?

Solution available with step-by-step explanation

SAT Circles Question 5 | Free SAT Prep | Aniko

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Question 5·Medium·Circles

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In the $xy$ -plane, the circle is defined by the equation

x^2 + y^2 - 6x + 8y + 9 = 0

Which of the following gives the center of the circle and its radius?

For circle equations on the SAT, aim to quickly convert from general form $x^2 + y^2 + Dx + Ey + F = 0$ to standard form $(x - h)^2 + (y - k)^2 = r^2$ by completing the square on the $x$ - and $y$ -terms. Group $x$ -terms and $y$ -terms, add the necessary constants to complete each square, and rewrite as binomial squares to read off $(h, k)$ and $r$ ; double-check signs (inside the parentheses are $x - h$ and $y - k$ ) and remember to take the square root of the constant on the right to get the radius, not the radius squared.

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Compare with the standard circle form

Write down the standard form of a circle, $(x - h)^2 + (y - k)^2 = r^2$ . How can you change the given equation so it looks like this?

Group x-terms and y-terms

Rearrange the equation so $x$ -terms are together and $y$ -terms are together. What expressions do you get for the $x$ and $y$ parts before completing the square?

Complete the square carefully

For each grouped expression, take half of the coefficient of $x$ (or $y$ ), square it, and add that value inside the group and to the other side of the equation. After that, can you rewrite each trinomial as a squared binomial?

Interpret signs and the radius correctly

Once you have an equation like $(x - h)^2 + (y - k)^2 = r^2$ , remember: the center is $(h, k)$ (notice the sign change inside the parentheses), and the radius is $r$ , not $r^2$ .

Desmos Guide

Graph the given circle

In Desmos, enter the equation x^2 + y^2 - 6x + 8y + 9 = 0 as one expression so you can see the circle on the coordinate plane.

Overlay a standard-form circle with sliders

Add a new expression (x - h)^2 + (y - k)^2 = r^2 and create sliders for h, k, and r. Adjust the sliders until this circle exactly overlaps the original one. The values of h and k at that point give the center, and the value of r gives the radius.

Step-by-step Explanation

Recall the standard form of a circle and group like terms

The standard form of a circle’s equation is

(x - h)^2 + (y - k)^2 = r^2

where the center is $(h, k)$ and the radius is $r$ .

Start with the given equation and group $x$ -terms together and $y$ -terms together:

x^2 + y^2 - 6x + 8y + 9 = 0

Reorder and group:

(x^2 - 6x) + (y^2 + 8y) + 9 = 0.

You will now complete the square for both the $x$ -group and the $y$ -group.

Complete the square for the x and y expressions

For each group, take half of the linear coefficient and square it:

For $x^2 - 6x$ : half of $-6$ is $-3$ , and $(-3)^2 = 9$ .
For $y^2 + 8y$ : half of $8$ is $4$ , and $4^2 = 16$ .

Add these squares inside the groups, and also to the other side of the equation to keep it balanced.

Start from

(x^2 - 6x) + (y^2 + 8y) + 9 = 0.

Add $9$ and $16$ inside the parentheses and to the right side:

(x^2 - 6x + 9) + (y^2 + 8y + 16) + 9 = 9 + 16.

Simplify the right-hand side:

(x^2 - 6x + 9) + (y^2 + 8y + 16) + 9 = 25.

Now subtract $9$ from both sides to isolate the completed-square groups:

(x^2 - 6x + 9) + (y^2 + 8y + 16) = 16.

Next, rewrite each trinomial as a squared binomial and compare to standard form.

Rewrite in standard form and identify center and radius

Each completed-square trinomial factors as a binomial squared:

$x^2 - 6x + 9 = (x - 3)^2$
$y^2 + 8y + 16 = (y + 4)^2$

So the equation becomes

(x - 3)^2 + (y + 4)^2 = 16.

Compare this to the standard form $(x - h)^2 + (y - k)^2 = r^2$ :

$(x - 3)^2$ means $h = 3$ .
$(y + 4)^2$ is the same as $(y - (-4))^2$ , so $k = -4$ .
$r^2 = 16$ , so $r = \sqrt{16} = 4$ .

Therefore, the correct choice is: The center is at $(3,-4)$ and the radius is $4$ . This corresponds to answer choice D.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation