SAT Circles Question 6 | Free SAT Prep | Aniko

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Question 6·Hard·Circles

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Which of the following equations represents the circle that passes through the points $(2,1)$ and $(6,1)$ and whose center lies on the line $y = x - 1$ ?

For circle questions in standard form on the SAT, first recall that $(x-h)^2 + (y-k)^2 = r^2$ has center $(h,k)$ and radius $r$ . Quickly read the center from each answer choice and use any given condition—like “the center lies on the line …” or a given chord—to eliminate options without doing full algebra. Once you have a likely center, plug in one of the given points to find or confirm $r^2$ , then match this to the remaining answer choice. This “read the center, apply conditions, then verify with one point” approach is much faster and less error-prone than expanding or doing heavy computations.

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Recall the circle equation format

Remember that a circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2 + (y-k)^2 = r^2$ . From each answer choice, you can read off a center $(h,k)$ and a value for $r^2$ .

Use the line condition first

The problem tells you the center lies on the line $y = x - 1$ . For each answer choice, plug the center $(h,k)$ into this line equation to see if it lies on the line.

Check the radius with a given point

Once you know which center could work, plug one of the given points, such as $(2,1)$ , into that circle’s equation to see if it satisfies the equation. That will give you the needed $r^2$ and help you confirm the correct choice.

Desmos Guide

Graph the given line and the points

Type y = x - 1 into Desmos to graph the line where the center must lie. Then add the points (2,1) and (6,1) using ({2,1}) and ({6,1}) so you can see the required locations on the coordinate plane.

Graph each answer choice as a circle

Enter each of the four equations from the answer choices into Desmos. For each circle you graph, look at its center (Desmos will show it or you can see it visually) and check whether that center lies on the line $y = x - 1$ and whether the circle passes exactly through both $(2,1)$ and $(6,1)$ .

Identify the circle that matches all conditions

Among the four circles, find the one whose center is on the line $y = x - 1$ and that goes through both plotted points. The equation of that circle is the correct answer.

Step-by-step Explanation

Use the standard form of a circle

The standard equation of a circle is

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

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

From each answer choice, you can read the center directly:

$(x+4)^2 + (y-3)^2 = 8$ has center $(-4,3)$ .
$(x-4)^2 + (y+3)^2 = 8$ has center $(4,-3)$ .
$(x-3)^2 + (y-4)^2 = 8$ has center $(3,4)$ .
$(x-4)^2 + (y-3)^2 = 8$ has center $(4,3)$ .

Use the condition that the center lies on the line

The center must lie on the line $y = x - 1$ . That means for the center $(h,k)$ we must have $k = h - 1$ .

Check each possible center from the answer choices:

For $(-4,3)$ : $h-1 = -4 - 1 = -5$ , but $k = 3$ , so it does not satisfy $y = x - 1$ .
For $(4,-3)$ : $h-1 = 4 - 1 = 3$ , but $k = -3$ , so it does not satisfy $y = x - 1$ .
For $(3,4)$ : $h-1 = 3 - 1 = 2$ , but $k = 4$ , so it does not satisfy $y = x - 1$ .
For $(4,3)$ : $h-1 = 4 - 1 = 3$ and $k = 3$ , so this center does lie on $y = x - 1$ .

So the center of the correct circle must be $(4,3)$ .

Use a point on the circle to find the radius

The circle passes through $(2,1)$ and $(6,1)$ . Use one of these points and the center $(4,3)$ to find the radius.

Compute the squared distance (which equals $r^2$ ) from $(4,3)$ to $(2,1)$ :

r^2 = (4-2)^2 + (3-1)^2 \\[0.7em] = 2^2 + 2^2 \\[0.7em] = 4 + 4 \\[0.7em] = 8

So the circle’s equation must have $r^2 = 8$ .

Write the equation and match it to a choice

Using the center $(4,3)$ and $r^2 = 8$ , the equation of the circle is

(x-4)^2 + (y-3)^2 = 8.

This matches choice D, so the correct answer is $(x-4)^2 + (y-3)^2 = 8$ .

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