Equal area properties for parabolas Consider the parabola y = x2. Let P, Q, and R be points on the parabola with R between P and Q on the curve. Let ℓP, ℓQ, and ℓR be the lines tangent to the parabola at P, Q, and R, respectively (see figure). Let P′ be the intersection point of ℓQ and ℓR, let Q′ be the intersection point of ℓP and ℓR, and let R′ be the intersection point of ℓP and ℓQ. Prove that Area ΔPQR = 2 · Area ΔP′Q′R′ in the following cases. (In fact, the property holds for any three points on any parabola.) (Source: Mathematics Magazine 81, 2, Apr 2008)
- a. P(−a, a2), Q(a, a2), and R(0, 0), where a is a positive real number
- b. P(−a, a2), Q(b, b2), and R(0, 0), where a and b are positive real numbers
- c. P(−a, a2), Q(b, b2), and R is any point between P and Q on the curve
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