Multiple choice

If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal, then which one of the following is correct ?

  1. $a,b$ and $c$ are in AP
  2. $a,b$ and $c$ are in GP
  3. $a,b$ and $c$ are in HP
  4. $a,b$ and $c$ do not follow any regular pattern
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, [b(c-a)]^2 - 4[a(b-c)][c(a-b)] = 0. This simplifies to b^2(c-a)^2 - 4ac(ab - b^2 - ac + bc) = 0. This is a known condition for a, b, c in HP.