Multiple choice

Equations ${ ax }^{ 2 }+bx+c=0$ and ${ cx }^{ 2 }+bx+a=0$have a common root and the difference of the other roots is 1. Maximum value of $\left| \frac { a }{ c } \right| $ is _______________.

  1. $\frac { \sqrt { 5 } +1 }{ 2 } $
  2. $\frac { \sqrt { 5 } -1 }{ 2 } $

  3. 1

  4. 2
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A Correct answer
Explanation

Let alpha be the common root. Then a*alpha^2 + b*alpha + c = 0 and c*alpha^2 + b*alpha + a = 0. Subtracting gives (a-c)(alpha^2 - 1) = 0. If a != c, alpha = 1 or -1. If alpha = 1, a+b+c=0. If alpha = -1, a-b+c=0. Using the difference of other roots = 1, the maximum value of |a/c| is (sqrt(5)+1)/2.

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