Question 612·Hard·Math

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In the $xy$ -plane, circle $C$ has equation $(x-3)^{2}+y^{2}=25$ . Circle $D$ has its center on the $x$ -axis, is tangent to the $y$ -axis, and is tangent to circle $C$ at exactly one point.

Which of the following is an equation of circle $D$ ?

A tangent circle touches another circle at exactly one point. When both centers lie on the $x$ -axis, their leftmost and rightmost points reveal where that touch can happen. Graph the given circle in Desmos and click its $x$ -axis crossings, then use tangency to the $y$ -axis to find the other circle’s radius. Don’t assume tangent circles must sit outside each other; one can sit inside the other.

Hints

Hint 1
The $x$ -axis passes through the given circle’s center, so its crossings with the circle are the leftmost and rightmost points. Graph the circle and $y=0$ in Desmos. What coordinates do you get?
Hint 2
A circle tangent to the $y$ -axis has a radius equal to its center’s horizontal distance from that axis. If its center is $(h,0)$ with $h>0$ , how far right does the circle reach?
Hint 3
For circles whose centers are on the same axis, internal tangency means one sits inside the other and their edges meet once. Compare their leftmost points first. If those differ, which ends could meet?

Step-by-step

Match the circles’ rightmost points
Step 1
Find circle C’s horizontal ends
Type the given circle equation and $y=0$ , the $x$ -axis, into Desmos. Click their two intersections: Desmos shows $(-2,0)$ and $(8,0)$ . Because the equation places $C$ ’s center on the $x$ -axis, these are its leftmost and rightmost points.
Step 2
Use tangency to the y-axis
Every offered equation puts $D$ ’s center to the right of the $y$ -axis, so call its center $(h,0)$ with $h>0$ . Its radius, the distance from center to edge, must equal the center’s distance to that axis. So $D$ ’s radius is $h$ .
Step 3
Find circle D’s horizontal ends
Move one radius left and right from $D$ ’s center to get its ends on the $x$ -axis:
$h-h=0,\qquad h+h=2h.$
Step 4
Match the only ends that can touch
The spans $-2$ to $8$ and $0$ to $2h$ overlap, so the circles cannot touch from the outside. And $C$ reaches left of $D$ , so $C$ cannot fit inside $D$ . For the circles to touch once, their rightmost points must match. Set those points’ $x$ -coordinates equal:
$2h=8.$
Step 5
Find D’s center and radius
Divide both sides by $2$ :
$h=4.$
So $D$ has center $(4,0)$ and radius $4$ .
Step 6
Write circle D’s equation
Use standard form, $(x-h)^2+(y-k)^2=r^2$ , with center $(4,0)$ and radius $4$ :
$(x-4)^2+y^2=4^2.$
Square the radius:
$(x-4)^2+y^2=16.$
This is the equation of circle $D$ . Choice B.

Question 612·Hard·Math

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