Question

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

Answer 1

Given,

$(1+m^2)x^2+2mcx+\left(c^{-}{a}^2\right)=0$

since the roots are equal

$\Rightarrow b^2=4ac$

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

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

$0=c^2-a^2-m^2{a}^2$

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

$\therefore c=a\sqrt{1+m^2}$