Question

If the equation $(1+m^2)x^2+2mcx+c^2-a^2=0$ has equal roots, then prove that $c^2=a^2(1+m^2)$.

Answer 1

Given equation is $(1+m^2)x^2+2mcx+c^2-a^2=0$

We need to prove $c^2=a^2(1+m^2)$

The roots are real and equal $\Delta =0$.

Therefore, $b^2-4ac=0$

$\Rightarrow (2mc{)}^2-4(1+m^2\left)\right(c^2-a^2)=0$

$\Rightarrow 4m^2{c}^2-4(c^2-a^2+m^2{c}^2-m^2{a}^2)=0$

$\Rightarrow m^2{c}^2-(c^2-a^2+m^2{c}^2-m^2{a}^2)=0$

$\Rightarrow m^2{c}^2-c^2+a^2-m^2{c}^2+m^2{a}^2=0$

$\Rightarrow c^2=a^2+a^2{m}^2$

$\Rightarrow c^2=a^2(1+m^2)$