AMC 10 · 2021 · #20
Grade 10 geometry-2dPick an answer.
Nothing here is numerically pinned down, so Tool #4 (Introduce a Variable) leads: name the wanted length d = FV and note that a parabola with a fixed vertex and axis is completely determined by that one number. Tool #1 (Draw a Diagram) puts V at the origin, F at (0,d), and the directrix at y=-d, which makes every distance in the problem readable from coordinates. Tool #13 (Convert to Algebra) then turns the focus-directrix definition into the equation x²=4dy and turns the two given lengths into two equations. Tool #16 (Change Focus) is the move that keeps the algebra clean: instead of measuring AF to the focus with a square root, measure the equal distance to the directrix, which is a plain subtraction. Two conditions on one point leave a single quadratic in d. Tool #3 (Eliminate Possibilities) tests whether both roots are real parabolas rather than algebraic ghosts, and Tool #9 (Solve an Easier Related Problem) finishes: the question wants the sum of the roots, which the coefficients already carry, so the ugly roots never have to be computed.
Name the focal distance
Put the vertex at the origin.
Once the vertex and axis are fixed, a parabola has exactly one shape parameter, so one unknown d describes every parabola the problem could mean.
10.G-GPE.A.1Introduce A VariableGet the equation from the definition
The definition gives the equation.
Squaring both distances erases the square root, and the matching y² and d² terms cancel, leaving only the clean relation x²=4dy.
Squaring both distances erases the square root, and the matching terms cancel to leave a clean relation.
▸ Why?
Every point of the curve is as far from the focus as from the directrix, so the two distances are equal.
▸ Why?
Squaring both sides of a true equation keeps it true, so the step loses nothing.
Measure to the directrix
Measure to the directrix instead.
A focal distance is secretly a vertical drop to a horizontal line, and vertical drops are just differences of y-coordinates.
9.A-CED.A.2Change Focus Count The ComplementWrite the other distance
Write the second distance with the formula.
Anchoring the vertex at the origin makes AV a plain distance formula, and the curve's own equation trades the unwanted x_A² away.
8.G.B.8Convert To AlgebraCollapse to a quadratic
It becomes a quadratic in the focal distance.
Two facts about the same point A — one per given length — are exactly enough to eliminate its coordinates and leave d standing alone.
9.A-CED.A.1Introduce A VariableCheck both roots count
Both roots land in the valid range.
Squaring can invent solutions that no longer fit the picture, so each root gets checked against the geometry before it is allowed into the total.
9.A-REI.B.4Eliminate PossibilitiesAdd without solving
The root relations give forty thirds.
A quadratic's coefficients already store the sum and product of its roots, so a question about the sum never needs the roots themselves.
9.A-SSE.A.1Solve An Easier Related ProblemWhen a question asks for the sum of all possible values, aim for one equation in that one unknown — the coefficients of a quadratic hand you the sum of its roots without you ever solving for them.
- Name the focal distance
- Write the parabola from its definition
- Measure to the directrix, not the focus
- Write AV with the distance formula
- Collapse it to a quadratic in d
- Check that both roots count
- Add the roots without finding them