Question 46·SAT Desmos Regression Practice·Geometry and Trigonometry

0 / 50 Questions completed

During a manufacturing test, the tip of a robotic arm rotates around a fixed pivot point. Three recorded positions of the tip are $(5,5)$ , $(-2,4)$ , and $(2,-4)$ in the $xy$ -plane.

Assuming these three points lie on a circle, which choice gives the coordinates of the pivot point (the center of the circle)?

The center of a circle is the same distance from every point on it. With three recorded positions but no given diameter, don't assume a pair's midpoint is the center. Fit a rearranged circle equation to the three points in Desmos. Halve the fitted $x$ and $y$ coefficients to recover the center.

Hints

Hint 1
A radius runs from the center to a point on the circle. All three positions must have the same radius from the pivot, so start with $(x-h)^2+(y-k)^2=r^2$ for a center $(h,k)$ .
Hint 2
For a Desmos regression, rewrite the circle equation as $x^2+y^2=ux+vy+w$ . The coefficients $u$ and $v$ are twice the center's coordinates.
Hint 3
Enter the positions in paired $x_1$ and $y_1$ columns, then fit $x_1^2+y_1^2\sim ux_1+vy_1+w$ . Desmos reports the coefficients under PARAMETERS. What must you do to $u$ and $v$ before reporting the pivot?

Step-by-step

Fit the circle through all three positions
Step 1
Turn equal distances into a circle equation
A radius is the distance from the center to any point on the circle. So every recorded position $(x,y)$ is the same distance $r$ from the unknown center $(h,k)$ . Squared horizontal and vertical changes add to $r^2$ :
$(x-h)^2+(y-k)^2=r^2$
Step 2
Rewrite the circle equation for fitting
Expand the squares:
$x^2-2hx+h^2+y^2-2ky+k^2=r^2$
Move the terms involving the center to the right:
$x^2+y^2=2hx+2ky+(r^2-h^2-k^2)$
Call the three right-side coefficients $u=2h$ , $v=2k$ , and $w=r^2-h^2-k^2$ . This gives Desmos the model $x^2+y^2=ux+vy+w$ . Don't use a pair's midpoint: that is the center only when the pair is known to be the ends of a diameter.
Step 3
Fit the model to the recorded positions
Enter the positions in the $x_1$ and $y_1$ table columns, keeping each pair in one row. Then type $x_1^2+y_1^2\sim ux_1+vy_1+w$ . The regression symbol $\sim$ tells Desmos to find coefficients that fit all three positions. Under PARAMETERS, Desmos shows $u=4$ , $v=2$ , and $w=20$ . These are coefficients, not yet the center's coordinates.
Step 4
Recover the pivot from the coefficients
Since $u=2h$ and $v=2k$ , halve both coefficients to get the center. Type $u/2$ and $v/2$ on new lines; Desmos prints $2$ and $1$ :
$h=\frac{u}{2}=\frac{4}{2}=2$ $k=\frac{v}{2}=\frac{2}{2}=1$
So the robotic arm's pivot is $(2,1)$ . Choice A.

Question 46·SAT Desmos Regression Practice·Geometry and Trigonometry

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this