A circle has its center at and passes through the point . Which of the following is an equation of this circle?

Solution available with step-by-step explanation

SAT Circles Question 48 | Free SAT Prep | Aniko

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Question 48·Medium·Circles

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A circle has its center at $(4,-2)$ and passes through the point $(10,1)$ . Which of the following is an equation of this circle?

For circle equation questions, immediately recall the standard form $(x-h)^2+(y-k)^2=r^2$ . First, use the center to fix the structure of the left side—this alone often lets you eliminate most choices by checking the signs inside the parentheses. Then, use the given point on the circle to compute $r^2$ via the distance formula (or directly as $(x-h)^2+(y-k)^2$ ), and match that value to the right side of the remaining options. This approach minimizes computation and quickly narrows to the correct equation.

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Think about the form of a circle's equation

Write down the general form of the equation of a circle with center $(h,k)$ and radius $r$ in terms of $x$ , $y$ , $h$ , $k$ , and $r$ .

Use the given center

Substitute $h=4$ and $k=-2$ into the standard circle equation. Be careful with the signs when $k$ is negative.

Use the point on the circle to find the radius

Use the distance from the center $(4,-2)$ to the point $(10,1)$ to find $r$ . You can plug this distance (squared) into the equation as $r^2$ .

Match with an answer choice

Once you know the left side structure from the center and the value of $r^2$ , look for the option whose left side uses that center and whose right side equals your $r^2$ .

Desmos Guide

Plot the given points

Enter the points (4,-2) and (10,1) in Desmos (for example, type (4,-2) on one line and (10,1) on another) so you can see the center and a point on the circle.

Graph each answer choice

Type each equation from the options into Desmos on separate lines, for example y=... is not needed; just enter them as implicit equations like (x-4)^2+(y+2)^2=45 so Desmos draws each circle.

Check the centers visually

For each circle, hover or click near its center to see its coordinates, and compare them to the point (4,-2). Eliminate any circles whose centers do not match (4,-2).

Check which circle passes through the point

For any circle(s) with the correct center, see whether the point (10,1) lies exactly on the circle. The correct equation will be the one whose graph includes both the center (4,-2) and the point (10,1) on the circle.

Step-by-step Explanation

Recall the standard form of a circle

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

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

So we need to identify $h$ , $k$ , and then $r^2$ from the information given.

Use the center to set up the circle equation

The problem says the center is $(4,-2)$ , so $h=4$ and $k=-2$ .

Substitute these into the standard form:

(x-4)^2 + (y-(-2))^2 = r^2,

which simplifies to

(x-4)^2 + (y+2)^2 = r^2.

Any correct answer must match this structure on the left side (same center).

Find the radius using the point on the circle

The circle passes through $(10,1)$ , which is a distance $r$ from the center $(4,-2)$ .

Use the distance formula for the radius, but keep it as $r^2$ (to match the equation form):

r^2 = (10-4)^2 + (1-(-2))^2.

Compute this expression to get the value of $r^2$ .

Write the full equation and match the choice

Evaluate

(10-4)^2 + (1-(-2))^2 = 6^2 + 3^2 = 36 + 9 = 45,

so $r^2 = 45$ .

Substitute into the equation from Step 2:

(x-4)^2 + (y+2)^2 = 45.

This matches answer choice D, so the correct equation is $(x-4)^2+(y+2)^2=45$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation