AMC 10 · 2022 · #22
Grade 11 algebrageometry-2dPick an answer.
The equation hides the roots, so name them. Writing the roots as a + bi and a - bi turns every one of the four vertices into a pair of plain coordinates, and Vieta turns the coefficients into two clean equations in a and b. Once the picture is coordinates, the area becomes a single product ab under the fixed budget a² + b² = 10, and that is a standard maximum.
The roots must be non-real
The discriminant must be negative.
Real roots would flatten the whole picture onto one line, and a flat quadrilateral has no area.
11.N-CN.C.7Eliminate PossibilitiesName the roots
The two roots are conjugates.
Two unknown complex numbers collapse into two real unknowns once you know they are mirror images across the real axis.
11.N-CN.A.2Introduce A VariableCompute the reciprocals
The reciprocals shrink toward the origin.
Because the product of the roots is 10, taking a reciprocal is the same as shrinking by 1/10 and flipping over the real axis.
11.N-CN.A.3Identify SubproblemsIdentify the shape
The four points form an isosceles trapezoid.
Conjugate pairs always come as up-down mirror twins, so the shape is forced to be symmetric across the horizontal axis.
10.G-GPE.B.4Draw A DiagramWrite the area
The area is proportional to one product.
Every length in the picture is a fixed multiple of a or b, so the area has to be a fixed multiple of their product.
10.G-GPE.B.7Identify SubproblemsMaximize it
The product peaks when the two are equal.
With the sum of two squares locked, a product is biggest when the two pieces are equal — the lopsided cases always lose.
With the sum of two squares locked, their product is biggest when the two pieces are equal.
▸ Why?
For a fixed total the parts multiply to the most when they are equal, and pulling them apart only costs.
▸ Why?
Every lopsided arrangement is beaten by its evened-out version, so only the balanced one can win.
Read off the value
The nearest listed value is 4.5.
Equal real and imaginary parts put each root at 45°, and doubling that real part is the whole of c.
11.N-RN.A.2Introduce A VariableConjugate roots make the four points mirror twins, the shape is forced to be a trapezoid whose area is just a constant times ab, and with a² + b² fixed that product is biggest when a = b.
- The roots must be non-real
- Name the roots a ± bi
- Reciprocals shrink toward the origin
- Plot: an isosceles trapezoid
- Area is a multiple of ab
- Maximize ab under a² + b² = 10
- Read off c