Question 1·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

0 / 25 Questions completed

In the $xy$ -plane, circle $C$ is defined by the equation $x^2+y^2-20x-8y+112=0$ . Circle $D$ is tangent to both coordinate axes and has its center in the first quadrant. Circle $D$ is externally tangent to circle $C$ , and the $x$ -coordinate of the center of circle $D$ is less than $10$ . What is the radius of circle $D$ ?

Your answer

An expanded circle equation hides its center and radius. Graph it in Desmos and click its top and bottom points to recover both. A circle tangent to both axes in the first quadrant has a center whose coordinates equal its radius. For external tangency, use the sum of the radii to relate the centers, then apply any bound to choose the valid solution.

Hints

Hint 1
The highest and lowest points of a circle are the ends of a vertical diameter, a line across the circle through its center. Graph circle $C$ as written. What do those two points tell you about its center and radius?
Hint 2
A circle tangent to an axis touches it at one point. Its radius equals its center’s distance from that axis. If $D$ touches both axes and its center is in the first quadrant, what must its center coordinates be?
Hint 3
Circles that touch from the outside are externally tangent: the distance between their centers equals the sum of their radii. Use the coordinate distance formula, then keep the solution whose center has $x$ -coordinate less than $10$ .

Step-by-step

Graph the first circle, then use center distance
Step 1
Find the top and bottom of circle C
Type the whole equation $x^2+y^2-20x-8y+112=0$ into Desmos, without adding $y=$ . Select the circle. Desmos marks its top at $(10,6)$ and bottom at $(10,2)$ . These points are the ends of a vertical diameter, which passes through the center.
Step 2
Read C’s center and radius
The center is halfway between the top height $6$ and bottom height $2$ , at height $4$ . Both points have $x$ -coordinate $10$ , so $C$ has center $(10,4)$ . Its radius, the distance from center to edge, is $6-4=2$ .
Step 3
Describe circle D with one number
Let $x$ be $D$ ’s radius. Because $D$ is tangent to each axis, its center is $x$ units from each axis. The center is in the first quadrant, where both coordinates are positive, so it must be $(x,x)$ . The given bound also says $0<x<10$ .
Step 4
Turn external tangency into an equation
For circles that touch from the outside, the distance between their centers equals the sum of their radii. Here the radii are $x$ and $2$ , so the center distance is $x+2$ , not their difference. The horizontal and vertical changes between $(x,x)$ and $(10,4)$ are $x-10$ and $x-4$ . Square the distance equation to write:
$(x-10)^2+(x-4)^2=(x+2)^2.$
Step 5
Solve using the center bound
Type the equation with $x_1$ in place of $x$ and $\sim$ in place of $=$ . Desmos uses $x_1$ as the number to find rather than as a graph coordinate. Add $\{0<x_1<10\}$ to enforce the given bound; without it, another radius also satisfies the tangency equation. Desmos reports $x_1=4$ under PARAMETERS. Since $x_1$ is $D$ ’s radius, the radius is $4$ . Grid in 4.

Question 1·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this