AMC 10 · 2006 · #19
Grade 10 geometry-2d
Pick an answer.
Tool #13 (Convert to Algebra) is the spine: the words "tangent" and "external" each translate into one exact equation about the distance from a center to the line, so the whole geometry problem becomes two equations in the two unknowns m and b. Tool #1 (Diagram) fixes which side of the line each circle is on, which is the only place the word "external" lives. Tool #4 (Introduce a Variable) handles that side information cleanly with a single sign symbol ε = ± 1 instead of splitting into a mess of absolute-value cases. Tool #7 (Subproblems) splits the work in the right order: subtracting the two equations kills b and leaves a one-variable equation for m, so find m first, then b. Tool #3 (Eliminate Possibilities) is needed at the end, because squaring an equation can invent roots that do not solve the original — every root has to be tested back, and one of the two survivors is ruled out only by the condition m ＞ 0.
Set up the picture and the unknowns
Write the line with two unknowns and translate each tangency.
An external tangent is the taut belt around the outside of two pulleys, so both circles stay on one side of it.
10.G-GPE.B.4Draw A DiagramTangent means distance equals radius
Tangency means the distance from a centre equals the radius.
Slide a line toward a circle: it misses, then just kisses it, then cuts it — the kissing moment is exactly when the perpendicular gap equals the radius.
10.G-C.A.2Convert To AlgebraWrite that distance in coordinates
A signed distance records which side each centre is on.
Going 1 across and m up covers √(1+m²) of line, so every vertical gap shrinks by exactly that factor when measured perpendicular.
8.G.B.7Convert To Algebra"External" is a statement about signs
Same side means a shared sign, removing the absolute values.
Same sign means same side of the line, and same side is exactly what the word "external" is asking for.
10.G-GPE.B.4Introduce A VariableSubtract to eliminate b
Subtracting the two conditions eliminates the intercept.
The intercept b shifts both centers the same way, so the difference of the two distances cannot see it at all.
The intercept shifts both centres the same way, so the difference of the two distances cannot see it at all.
▸ Why?
Shifting two quantities by the same amount leaves their difference exactly as it was.
▸ Why?
Subtracting one true equation from another keeps the result true, so the elimination is legitimate.
Solve for the slope, then test the roots
Two roots appear and both are genuine, but only one has positive slope.
Two roots, two real tangents: the flat one is the x-axis, and m ＞ 0 is exactly the clue that sends you to the other.
9.A-REI.B.4Eliminate PossibilitiesPut the slope back to get b
Substituting back gives 912/119, choice (E).
One equation gave b; the unused second circle then confirms it for free.
9.A-REI.B.3Convert To Algebra"Tangent" means the distance from the center to the line equals the radius, and "external" means both centers land on the same side of the line — write those two facts as equations, subtract them to make b disappear, and the slope 120/119 falls out and hands you b = 912/119.
- Set up the picture and the unknowns
- Tangent means distance equals radius
- Write that distance in coordinates
- "External" is a statement about signs
- Subtract to eliminate b
- Solve for the slope, then test the roots
- Put the slope back to get b