2017 · Q22
If the roots of the equation a(b-c)x²+b(c-a)x+c(a-b)=0 are equal, then which one of the following is correct ?
Elementary MathematicsAlgebra
- A2b = a + c
- Bb²=ac
- C2/b = 1/a + 1/c
- D1/b = 1/a + 1/c
Explanation
Substituting x = 1: a(b−c)+b(c−a)+c(a−b) = 0, so x = 1 is always a root. Equal roots mean the double root is 1, so the product of the roots c(a−b)/[a(b−c)] = 1. This gives c(a−b) = a(b−c), i.e. ac−bc = ab−ac, or 2ac = b(a+c), which rearranges to 2/b = 1/a+1/c. Hence option (C).