Question

If the equation (1+ m)2x2+2mx+c2-a2=0 has equal roots, show that c2=a2(1+m)2

Answer 1

Given Equation is (1 + m^2) x^2 + 2mx + c^2 - a^2 = 0

Given that the equation has equal roots.

We know that when the roots are real and equal, then the quadratic equation b^2 - 4ac = 0.

Here b = 2mc, c = (c^2 - a^2), a = (1 + m^2).

Now,

b^2 - 4ac = 0

(2mc)^2 - 4(1 + m^2)(c^2 - a^2) = 0

4m^2c^2 - 4(c^2 - a^2 + m^2c^2 - m^2a^2) = 0

4m^2c^2 - 4c^2 + 4a^2 - 4m^2c^2 + 4m^2a^2 = 0

4m^2a^2 - 4c^2 + 4a^2 = 0

m^2a^2 - c^2 + a^2 = 0

a^2(m^2 + 1) - c^2 = 0

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


LHS = RHS.


Hope this helps!

Answer 2

Hi ,

Compare

( 1 + m )² x² + 2mx + c² - a² = 0 with

Ax² + bx + C = 0 ,

A = ( 1 + m)² ,

b = 2m ,

C = c² - a² ,

According to the problem given ,

Roots are equal .

Therefore ,

b² = 4AC

( 2m )² = 4 ( 1 + m )² ( c² - a² )

4m² = 4( 1 + m)² ( c² - a² )

m² = ( 1 + m )² ( c² - a² )

m² = c² ( 1 + m)² - a² ( 1 + m )²

a² ( 1 + m )² = c² ( 1 + m )² - m²

I hope this helps you.

:)