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?

Concept:

For any quadratic equation, ax² + bx + c = 0. We have discriminant, D = b² - 4ac, then the given quadratic equation has:

I. Distinct and real roots if D > 0.

II. Real and repeated roots, if D = 0.

III. Complex roots and conjugate of each other, D < 0.

If a, b and c are in HP, then the harmonic mean, \(b = \frac{{2ac}}{{a + c}}\)

Calculation:

Given: a (b - c) x² + b (c - a) x + c (a - b) = 0 has equal roots ⇒ Discriminant, D = 0.

By comparing the given equation with the quadratic equation, ax² + bx + c = 0. We get, a’ = a (b - c) , b’ = b (c - a) and c’ = c (a - b)

As, Discriminant, D = 0.

⇒ D = b² - 4ac = b² × (c - a) ² - 4 × a (b - c) × c (a - b) = 0

⇒ D = (bc + ab - 2ac) ² = 0

⇒ bc + ab - 2ac = 0

⇒ b × (a + c) = 2ac

\(\Rightarrow b=\frac{2ac}{a+c}\)