AMC 10 · 2021 · #21
Grade 11 geometry-2dPick an answer.
A general conic carries five unknown coefficients, which is exactly as many as we have points — solvable, but ugly. Tool #1 (Draw a Diagram) cuts that down first: plotting the five roots shows they come in mirror pairs across the x-axis, and since the ellipse through them is unique, the ellipse must have that same mirror symmetry. That single observation kills the xy and y terms and puts the center on the x-axis. Tool #4 (Introduce a Variable) then names what is left — the center (h,0) and the two squared axis lengths a² and b² — so three unknowns absorb all five points. Tool #7 (Identify Subproblems) keeps the finish orderly: solve the quadratics first, then h, then a² and b², then the eccentricity, one small job at a time.
Solve for the five roots
Read the roots off each factor.
A product is zero exactly when one factor is zero, so a factored equation is really three tiny equations stacked in a row.
A product is zero exactly when one factor is zero, so a factored equation is really several tiny equations.
▸ Why?
Two nonzero numbers can never multiply to zero, so no root can hide outside the factors.
▸ Why?
The factors' solution sets cover everything without overlapping, so the roots can simply be listed together.
Read the symmetry
The points are symmetric about the axis.
If the points look the same in a mirror and only one curve fits them, the curve has to look the same in that mirror too.
10.G-GPE.B.4Draw A DiagramUse the point on the axis
That point fixes one axis.
A point sitting on the center's own horizontal line is a vertex, and a vertex hands you the semi-axis for free.
9.A-CED.A.2Introduce A VariableFind the centre
The other points fix the centre.
Two equations that both equal b² can be glued to each other, and gluing them throws the unknown you did not want off the page.
9.A-REI.B.3Introduce A VariableRead both axis lengths
Compute both axis lengths.
Once the center is pinned down, every remaining length is just a distance from that center.
9.A-REI.B.3Identify SubproblemsConvert to eccentricity
Converting and adding gives 7.
Eccentricity only cares about the ratio of the axes, so the messy common factor in a² and b² is guaranteed to cancel.
10.G-GPE.B.4Introduce A VariableWhen the points you must fit are mirror images of each other and only one curve fits them, the curve is forced to be a mirror image too — that free symmetry is what turns five unknowns into one.
- Solve for the five roots
- Plot the points and read the symmetry
- Use the point on the axis
- Eliminate the axis lengths to find h
- Read off both axis lengths
- Convert axes into eccentricity