If the equation (1+m^2 ) x^2+2 m c x+ (c^2-a^2 )=0 has equal root… - Vidyadip

Question

If the equation $\left(1+m^2\right) x^2+2 m c x+\left(c^2-a^2\right)=0$ has equal roots, prove that $c^2=a^2\left(1+m^2\right)$

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Answer

Here roots are equal,
$
\therefore D=B^2-4 A C=0
$
Here, $A=1+m^2, B=2 m c, C=\left(c^2-a^2\right)$
$
\therefore(2 m c)^2-4\left(1+m^2\right)\left(c^2-a^2\right)=0
$
or, $4 m^2 c^2-4\left(1+m^2\right)\left(c^2-a^2\right)=0$
or, $m^2 c^2-\left(c^2-a^2+m^2 c^2-m^2 n^2\right)=0$
or, $m^2 c^2-c^2+a^2-m^2 c^2+m^2 a^2=0$
or, $-c^2+a^2+m^2 a^2=0$
or, $c^2=a^2\left(1+m^2\right)$
Hence Proved.

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