Question

if the quadratic equation(1+m^2)x^2+2mcx+c^2-a^2=0 has equal roots prove that c^2=a^2(1+m^2)

Answer 1

Quadratic Equation = ( 1 + m² ) x² + ( 2 mc ) x + c² - a² = 0

Coefficients of x² ( a ) = ( 1 + m² )

Coefficients of x ( b ) = ( 2 mc )

Coefficients of constant ( c ) = c² - a²

Discriminant = b² - 4ac = 0 ( Given )

( 2mc )² - 4 ( 1 + m² ) ( c² - a² ) = 0

4 m² c² - 4 ( c² - a² + m².c² - m².a² ) = 0

4 m².c² - 4 c² + 4 a² - 4 m².c² + 4 m².a² = 0

4 ( m².c² - c² + a² - m².c² + m².a² ) = 0

m².c² - c² + a² - m².c² + m².a² = 0 / 4 = 0

m².c² gets cancelled.

a² + m².a² - c² = 0

Taking ' a ' as a common factor:

a² ( 1 + m² ) - c² = 0

a² ( 1 + m² ) = c²

Hence Proved