Multiple choice

If a root of the equation $a_1 x^2 + b_1 x + c_1 = 0 $ is the reciprocal of a root of the equation $a_2 x^2 + b_2 x + c_2 = 0 $ then

  1. $(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) $
  2. $(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) $
  3. $(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) $
  4. $(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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A Correct answer
Explanation

If alpha is a root of the first, 1/alpha is a root of the second. Substituting 1/x for x in the second equation gives a2(1/x)^2 + b2(1/x) + c2 = 0, or c2*x^2 + b2*x + a2 = 0. Comparing with a1*x^2 + b1*x + c1 = 0, we get a1/c2 = b1/b2 = c1/a2. This leads to the condition (a1*a2 - c1*c2)^2 = (a1*b2 - b1*c2)(a2*b1 - b2*c1).

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