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

$(1 + m^{2}) x^{2} + 2 m c x + c^{2} - a^{2} = 0$ has equal roots

⇒ $b^{2} - 4 a c$ = 0

⇒ $(2 m c)^{2} - 4 (1 + m^{2}) (c^{2} - a^{2})$ = 0

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

⇒ $4 m^{2} c^{2} - 4 c^{2} + 4 a^{2} - 4 m^{2} c^{2} + 4 m^{2} a^{2}$ = 0

⇒ $4 m^{2} a^{2} - 4 c^{2} + 4 a^{2}$ = 0

⇒ $m^{2} a^{2} - c^{2} + a^{2}$ = 0

⇒ $a^{2} (1 + m^{2}) - c^{2}$ = 0

⇒ $c^{2}$ = $a^{2} (1 + m^{2})$
hence proved.

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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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