If a root of the equation is the reciprocal of a root of the equation

MATHEMATICS · Single correct · AoPS · Mathematics · Quadratic Equations - Part 1 · Quadratic Equations - Part 1

Problem

If a root of the equation a1x2+b1x+c1=0a_1 x^2 + b_1 x + c_1 = 0 is the reciprocal of a root of the equation a2x2+b2x+c2=0a_2 x^2 + b_2 x + c_2 = 0, then: (a) (a1a2−c1c2)2=(a1b2−b1c2)(a2b1−b2c1)(a_1 a_2 - c_1 c_2)^2 = (a_1 b_2 - b_1 c_2)(a_2 b_1 - b_2 c_1) (b) (a1a2−b1b2)2=(a1b2−b1c2)(a2b1−b2c1)(a_1 a_2 - b_1 b_2)^2 = (a_1 b_2 - b_1 c_2)(a_2 b_1 - b_2 c_1) (c) (b1c2−b2c1)2=(a1b2−b1c2)(a2b1+b2c1)(b_1 c_2 - b_2 c_1)^2 = (a_1 b_2 - b_1 c_2)(a_2 b_1 + b_2 c_1) (d) (b1c2−b2c1)2=(a1b2+b1c2)(a2b1−b2c1)(b_1 c_2 - b_2 c_1)^2 = (a_1 b_2 + b_1 c_2)(a_2 b_1 - b_2 c_1)

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