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

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Solution

$(1 + m^{2}) x^{2} + (2 m c) x + (c^{2} - a^{2}) = 0$
$B^{2} - 4 A C = 0 ⟹ 4 m^{2} c^{2} - 4 (c^{2} - a^{2}) (1 + m^{2}) = 0$
$c^{2} = a^{2} (1 + m^{2})$

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