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

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Solution

Given :-
$(H m^{2}) x^{2} + 2 c m x + (c^{2} - a^{2}) = 0$ has equal roots.
$\Rightarrow B^{2} = 4 A C$
$\Rightarrow (2 c m)^{2} = 4 (1 + m^{2}) (c^{2} - a^{2})$
$\Rightarrow c^{2} m^{2} = (1 + m^{2}) (c^{2} - a^{2})$
$\Rightarrow c^{2} m^{2} = c^{2} - a^{2} + C^{2} m^{2} - a^{2} m^{2}$
$\Rightarrow c^{2} = a^{2} (1 + m^{2})$

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