AMC 10 · 2023 · #10
Grade 10 geometry-2dPick an answer.
The problem is handed over in words about tangency and axes, so Tool #13 (Convert to Algebra) owns it: once each circle becomes an equation, everything after that is symbol pushing. Tool #1 (Draw a Diagram) does the first real work, because "tangent to the y-axis at the origin" is a picture statement, and reading the picture is what pins each center to a single point rather than a range. Tool #15 (Organize Information in More Ways) is the hinge — the standard form (x-h)²+(y-k)²=r² hides what is useful here, and expanding both circles into the form x²+y²=something linear lines the two equations up so that their curved parts are literally identical. Tool #16 (Change Focus) is the punchline: the question asks about a line through two points, and the reflex is to hunt down both points first. Chasing the line itself instead makes the two points unnecessary, and the whole problem collapses to one subtraction. Tool #7 (Identify Subproblems) then handles a small side job at the end: confirming that the two circles really do meet twice, so the line found by subtraction is a line through actual points.
Pin down both centres
Tangency fixes each centre.
A tangent line is exactly one radius away from the center, so saying "tangent to that axis" is the same as saying "the center is r units off that axis."
10.G-CO.A.1Draw A DiagramWrite each equation
Write each circle as an equation.
The circle equation is just the distance formula with both sides squared, so "lies on the circle" becomes an equation you can do algebra to.
10.G-GPE.A.1Convert To AlgebraExpanding kills the constants
Expanding kills the constant terms.
Rewriting both circles so the curved part x²+y² stands alone on the same side makes the two equations directly comparable instead of merely similar.
9.A-SSE.A.2Organize Information In More WaysSubtract to get the line
Subtracting makes the squares vanish.
Both circles carry the same curved piece x²+y², so subtracting one from the other destroys all the curvature and leaves behind exactly the straight line through their crossing points.
Both circles carry the same curved piece, so subtracting one from the other destroys the curvature.
▸ Why?
Subtracting two quantities that share the identical block removes that block entirely.
▸ Why?
Each circle's equation says a point sits one fixed distance from a centre, which is where that block comes from.
Find the crossings
Substitute to find both crossings.
Feeding the line back into one circle turns a two-variable system into a single quadratic, and its two roots are the two crossing points.
9.A-REI.B.4Identify SubproblemsRead the slope
The slope is two fifths.
Once a line is written as y=mx+b, the number multiplying x is the slope, with nothing left to compute.
8.F.A.3Convert To AlgebraTwo circle equations both contain the same x²+y², so subtracting one from the other wipes out the curvature and leaves the straight line through their two crossing points.
- Pin down both centers
- Write each circle's equation
- Expand; the constants cancel
- Subtract to get the line
- Find the two crossing points
- Read the slope