Question 704·Hard·Math

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A circle with center $(1,-4)$ has area $36\pi$ square units. The circle is dilated by a scale factor of $\tfrac12$ about its center and then translated 5 units to the right and 2 units up in the $xy$ -plane. Which of the following equations represents the image of the circle after these transformations?

A circle’s area gives its radius through $A=\pi r^2$ . A dilation about the center scales the radius but leaves the center in place; a translation then moves the center without changing that new radius. Use a Desmos regression to find the positive radius from the area. Finally, put the new center and radius into $(x-h)^2+(y-k)^2=r^2$ , watching the signs inside the parentheses.

Hints

Hint 1
The area formula is $A=\pi r^2$ . Substitute the given area, then find the positive radius; a distance from the center to the circle can’t be negative.
Hint 2
A dilation about a circle’s center changes the radius but doesn’t move the center. Multiply the original radius by the scale factor before you handle the translation.
Hint 3
A translation moves the center right or left and up or down. Move the center’s coordinates first; then use $(x-h)^2+(y-k)^2=r^2$ . The signs inside the parentheses are opposite the center’s coordinates.

Step-by-step

Find the radius, then transform the circle
Step 1
Turn the area into an equation
The given $36\pi$ square units is the area, the space inside the circle. The area formula is $A=\pi r^2$ , where $r$ is the radius, or distance from the center to the circle. Substitute the given area:
$36\pi=\pi r^2.$
Step 2
Find the original radius
Type $36\pi\sim\pi r^2\{r>0\}$ in Desmos. The $\sim$ asks Desmos to find $r$ , and the restriction keeps the positive solution because a radius can’t be negative. Under PARAMETERS, Desmos shows $r=6$ , so the original radius is $6$ units.
Step 3
Scale the radius
A dilation multiplies distances from its center by the scale factor. Because this dilation is about the circle’s center, that center stays at $(1,-4)$ , while the radius becomes $6\cdot\tfrac12=3$ . Scale the radius, not the center.
Step 4
Translate the center
A translation slides the circle without changing its radius. Add $5$ to the center’s $x$ -coordinate to move right, and add $2$ to its $y$ -coordinate to move up:
$(1+5,-4+2)=(6,-2).$
So the new center is $(6,-2)$ , and the radius is still $3$ .
Step 5
Write the image’s equation
In standard form, $(x-h)^2+(y-k)^2=r^2$ , the center is $(h,k)$ and the right side is the radius squared. Substitute the new center $(6,-2)$ and radius $3$ :
$(x-6)^2+(y-(-2))^2=3^2.$
Simplify the sign:
$(x-6)^2+(y+2)^2=3^2.$
Square the radius:
$(x-6)^2+(y+2)^2=9.$
The image is the circle centered at $(6,-2)$ with radius $3$ . Choice C.

Question 704·Hard·Math

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