Question

If the equation (1+m²)n²x²+2mncx +(c²-a²)=0 of x has equal roots, prove that c²=a²(1+m²)

Answer 1

Formula :

Let us take a quadratic equation

                      ax² + bx + c = 0

where a, b, c are constants with non-zero a

Then, the equation will be equal roots, only when the value of the discriminant be 0

    i.e., b² - 4ac = 0

Solution :

The given quadratic equation is

  (1 + m²)n²x² + 2mncx + (c² - a²) = 0

For equal roots, D = 0

⇒ (2mnc)² - 4 (1 + m²)n² (c² - a²) = 0

⇒ 4m²n²c² - 4 (n² + m²n²) (c² - a²) = 0

⇒ 4m²n²c² - 4n²c² + 4n²a² - 4m²n²c² + 4m²n²a² = 0

⇒ - 4n²c² + 4n²a² + 4m²n²a² = 0

⇒ c² = a² + m²a²

⇒ c² = a² (1 + m²)

which is the required condition for equal roots.

Answer 2

hope it will help you