Question

if the eq
$(1 + m {}^{2} )x {}^{2} + 2mcx + (c {}^{2} - a {}^{2} ) = 0$
has equal roots then prove that
$c {}^{2} = a {}^{2} (1 + m {}^{2} )$

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quadratic equation ​

Answer 1

Answer:

Step-by-step explanation:

Given :

The equation $(1+m^2)x^2+2mcx+(c^2-a^2)=0$

has 2 equal roots,

then prove : $c^2 = a^2(1+m^2)$

Solution :

We know that,

for quadratic equation with equal roots,

$b^2 - 4ac = 0$

Proof :

⇒ $(2mc)^2 - 4(1+m^2)(c^2-a^2) = 0$

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

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

⇒ $- 4c^2 + 4a^2 +4 m^2a^2 = 0$

⇒ $4(c^2 - a^2 + m^2a^2) = 0$

⇒ $c^2 - a^2 + m^2a^2 = 0$

⇒ $c^2 - a^2(1 + m^2) = 0$

⇒ $c^2 = a^2(1 + m^2)$

Hence proved,.