SAT Circles Question 26 | Free SAT Prep | Aniko

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Question 26·Hard·Circles

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A circle with center $(1,-4)$ has area $36\pi$ square units. The circle is dilated by a scale factor of $\tfrac12$ about its center and then translated 5 units to the right and 2 units up in the $xy$ -plane. Which of the following equations represents the image of the circle after these transformations?

For transformation questions with circles, first use $A = \pi r^2$ to get the radius from the area. Then apply transformations in order: a dilation about the center only changes the radius (multiply the radius by the scale factor), while a translation only changes the center coordinates (add/subtract from $x$ and $y$ as directed). Finally, plug the new center and radius into $(x-h)^2 + (y-k)^2 = r^2$ , watching sign conventions closely so you don’t mix up left/right or up/down in the equation.

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Connect area to radius

Use the circle area formula $A = \pi r^2$ and the given area $36\pi$ to find the original radius of the circle.

Think about what dilation changes

A dilation by a scale factor of $\tfrac12$ about the center changes all distances from the center (including the radius) by that factor. How does this affect the radius?

Track how the center moves

Dilation about the center does not move the center. After you know the center and radius of the dilated circle, adjust the center’s coordinates for a translation 5 units right and 2 units up.

Use the standard circle equation

Once you know the new center $(h,k)$ and the new radius $r$ , plug into $(x - h)^2 + (y - k)^2 = r^2$ . Be careful with the signs for $k$ if the $y$ -coordinate is negative.

Desmos Guide

Confirm the new radius numerically

In Desmos, type 6*(1/2) to verify that dilating the original radius 6 by a factor of $\tfrac12$ gives a new radius of 3.

Graph each answer choice

Enter each option exactly as written:

(x-6)^2 + (y+2)^2 = 9
(x+6)^2 + (y-2)^2 = 9
(x-6)^2 + (y+2)^2 = 36
(x+4)^2 + (y-6)^2 = 9 Desmos will draw four circles on the same axes.

Plot the expected center and check radius

Add the point (6,-2) in Desmos. Then look at which circle is centered at this point and has radius 3 (for example, check that points like (9,-2) or (6,1) lie on that circle). The equation of that circle matches the correct choice.

Step-by-step Explanation

Find the original radius from the area

The area of a circle is given by $A = \theta r^2$ , where $\theta$ here is $\tau$ ? Wait — that's not right. The correct formula is:

A = \pi r^2

We are told $A = 36\pi$ , so set up:

\pi r^2 = 36\pi

Divide both sides by $\pi$ :

r^2 = 36

So the original radius is $r = 6$ .

Apply the dilation to the radius

A dilation with scale factor $\tfrac12$ about the center multiplies all distances from the center by $\tfrac12$ .

The center stays at $(1,-4)$ .
The radius gets multiplied by $\tfrac12$ .

So the new radius is:

6 \cdot \frac12 = 3

After the dilation, the circle has center $(1,-4)$ and radius $3$ .

Apply the translation to the center

Now translate the dilated circle 5 units to the right and 2 units up.

Moving 5 units right adds 5 to the $x$ -coordinate: $1 + 5 = 6$ .
Moving 2 units up adds 2 to the $y$ -coordinate: $-4 + 2 = -2$ .

So after the translation, the image circle has:

Center at $(6,-2)$
Radius $3$ (unchanged by translation).

Write the equation of the image circle

The standard form of a circle with center $(h,k)$ and radius $r$ is

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

We have $h = 6$ , $k = -2$ , and $r = 3$ , so $r^2 = 9$ .

Substitute these values:

(x - 6)^2 + (y - (-2))^2 = 9

which simplifies to

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

So the correct equation is $(x-6)^{2} + (y+2)^{2} = 9$ .

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