Question 49·SAT Desmos Regression Practice·Geometry and Trigonometry

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During a drone light show, a circular safety zone is drawn on a coordinate grid so that its boundary passes through the three GPS markers $(2,1)$ , $(6,5)$ , and $(0,4)$ .

In the $xy$ -plane, the boundary of the safety zone can be modeled by the equation

x^2+y^2+ax+by+c=0,

where $a$ , $b$ , and $c$ are constants.

Which choice is the value of $c$ ?

Three points on a circle can determine its circle equation, even when its center isn’t given. If you need one coefficient, put the points in a Desmos table and fit the equation with a regression. Keep track of which side holds the constant: moving it across the equals sign changes its sign.

Hints

Hint 1
A point is on a circle when its coordinates make the circle’s equation true. The three markers give three conditions for the unknown coefficients. How could you enter the points together in Desmos?
Hint 2
In a regression, Desmos can fit $x_1^2+y_1^2\sim ux_1+vy_1+w$ to the three points. The separate coefficients $u$ , $v$ , and $w$ stand in for the unknown numbers in the equation.
Hint 3
The constant term is the number with no $x$ or $y$ attached. The fitted equation puts that number on the right, but the question puts $c$ on the left. What happens to its sign when it changes sides?

Step-by-step

Fit the circle’s coefficients in Desmos
Step 1
Rewrite the equation for a regression
To fit the three unknown coefficients, move them to the right side of the given circle equation. Subtract $ax+by+c$ from both sides:
$x^2+y^2=-ax-by-c.$
Name the right-side coefficients $u=-a$ , $v=-b$ , and $w=-c$ :
$x^2+y^2=ux+vy+w.$
Keep $w=-c$ in view.
Step 2
Enter the boundary points
Each marker is on the boundary, so its coordinates satisfy the same equation. Enter all three points in a Desmos table, keeping each point’s $x$ - and $y$ -coordinates in the same row: $x_1$ gets $2,6,0$ , and $y_1$ gets $1,5,4$ . The regression will use these rows.
Step 3
Fit the right-side coefficients
Type $x_1^2+y_1^2\sim ux_1+vy_1+w$ . The regression uses $\sim$ to find values that match the table rows. Under PARAMETERS, Desmos shows $u=6.2$ , $v=7.8$ , and $w=-15.2$ . That last value is the right-side constant, not $c$ .
Step 4
Find the requested constant
Since $w=-c$ , the constant changes sign when it moves back to the left. Type $-w$ on a new Desmos line. It prints $15.2$ ; tap the fraction button to see $\frac{76}{5}$ . So the constant term in the given equation is $c=\frac{76}{5}$ . Choice C.

Question 49·SAT Desmos Regression Practice·Geometry and Trigonometry

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