Question

If the quadratic equation (a² - b²) x² + (b² - c²) x + c² - a² = 0 has equal roots, then which of the following is true:
1. b² = c² + 2a²
2. b² + c² = 2a²
3. b² - c² = 2a²
4. b² + c² = a²

Answer 1

Correct Answer - Option 2 : b² + c² = 2a²

Concept:

If the quadratic equation is given by Ax² + Bx + C = 0.

Then,

Sum of roots = - \(\frac{B}{A}\)

Product of roots = \(\frac{C}{A}\)

Given:

(a2 - b2) x2 + (b2 - c2) x + c2 - a2 = 0

Where,

A = (a2 - b2), B = (b2 - c2), C = c2 - a2

Calculation:

Here,

A + B + C = (a2 - b2) + (b2 - c2) + (c2 - a2) = 0.

So, we can say that x = 1 is a root of a given quadratic equation. According to question both roots are same.

Then,

Roots (x) = 1, 1

Product of Roots = \(\frac{C}{A}\) 

1 × 1 = \(\frac{(c^2~-~a^2)}{(a^2~-~b^2)}\)

(a2 - b2) = (c2 - a2)

By arranging, we get

b² + c² = 2a²