Question

If the equation \[\left(1 + m^2 \right) x^2 + 2mcx + \left(c^2 - a^2 \right) = 0\] has equal roots, prove that c² = a²(1 + m²).

Answer 1

The given equation \[\left(1 + m^2 \right) x^2 + 2mcx + \left(c^2 - a^2 \right) = 0\], has equal roots

Then prove that`c^2 = (1 + m^2)`.

Here,

`a = (1 + m^2), b = 2mc and c = (c^2 - a^2)`

As we know that `D = b^2 - 4ac`

Putting the value of `a = (1 + m^2), b = 2mc and c = (c^2 -  a^2)`

`D = b^2 - 4ac`

` = {2mc}^2 - 4xx (1 +m^2) xx (c^2 - a^2)`

` = 4 (m^2 c^2) - 4(c^2 -a^2 + m^2c^2 - m^2 a^2)`

` = 4m^2c^2 - 4c^2 + 4a^2 - 4m^2 c^2 + 4m^2a^2`

` = 4a^2 + 4m^2 a^2 = 4c^2`

The given equation will have real roots, if D  = 0

 `4a^2 + 4m^2 a^2 - 4c^2 = 0`

`4a^2 + 4m^2a^2 = 4c^2`

`4a^2 + (1 + m^2 ) = 4c^2`

`a^2 (1 +m^2) = c^2`

Hence, `c^2 = a^2 (1 + m^2)`.