Question 48·SAT Desmos Regression Practice·Geometry and Trigonometry

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In the coordinate plane, circle $C$ passes through points $P$ , $Q$ , and $R$ shown in the figure.

Which equation represents circle $C$ ?

Three non-collinear points on a circle determine one circle, even when its center isn't marked. A circle regression lets you enter the points in a Desmos table, fit an expanded equation, and convert it to center-radius form. Watch the squared radius on the right side: it includes both the horizontal and vertical distances from the center.

Hints

Hint 1
Each dot gives an ordered pair: read its horizontal coordinate first and its vertical coordinate second. Keep the $x$ - and $y$ -coordinates together as you enter the three points in a Desmos table.
Hint 2
A regression finds constants that fit the table. Use $x_1^2+y_1^2\sim ux_1+vy_1+w$ to find $u$ , $v$ , and $w$ . The $\sim$ symbol tells Desmos to fit the equation rather than graph it.
Hint 3
In center-radius form, $(x-h)^2+(y-k)^2=r^2$ , the center is $(h,k)$ . The fitted coefficients satisfy $u=2h$ and $v=2k$ , so halve both. Then $r^2=w+h^2+k^2$ gives the right side.

Step-by-step

Fit the circle through the three points
Step 1
Enter the points from the figure
Read each labeled dot against the grid: $P=(-1,2)$ , $Q=(3,6)$ , and $R=(5,0)$ . Enter them as rows in a Desmos table. Keep each point's two coordinates in the same row so the fit uses the points actually shown.
Step 2
Fit an expanded circle equation
Type $x_1^2+y_1^2\sim ux_1+vy_1+w$ . Here $x_1$ and $y_1$ name the table columns, and the regression symbol $\sim$ asks Desmos to fit all three points. Under PARAMETERS, Desmos shows $u=5$ , $v=5$ , and $w=0$ .
Step 3
Turn the fitted coefficients into the center
The center-radius form is $(x-h)^2+(y-k)^2=r^2$ , where $(h,k)$ is the center. Expand the squares:
$x^2-2hx+h^2+y^2-2ky+k^2=r^2.$
Move the terms containing the center to the right:
$x^2+y^2=2hx+2ky+(r^2-h^2-k^2).$
So $u$ and $v$ are twice the center coordinates, not the coordinates themselves. Type $h=u/2$ and $k=v/2$ . Desmos shows $h=k=2.5$ , exactly $\frac52$ .
Step 4
Find the squared radius
The expanded equation shows $w=r^2-h^2-k^2$ . Add $h^2+k^2$ to both sides:
$r^2=w+h^2+k^2.$
Type $q=w+h^2+k^2$ . Desmos gives $q=12.5$ ; its fraction button shows $\frac{25}{2}$ . This is the squared radius, which is what the circle equation needs.
Step 5
Write the equation of circle C
Put the center $\left(\frac52,\frac52\right)$ and $r^2=\frac{25}{2}$ into $(x-h)^2+(y-k)^2=r^2$ :
$(x-\frac52)^2+(y-\frac52)^2=\frac{25}{2}.$
The minus signs place the center at positive coordinates. This is the equation of circle $C$ . Choice A.

Question 48·SAT Desmos Regression Practice·Geometry and Trigonometry

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