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

Question

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

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Answer

Given:
$(1 + m^2)x^2 + 2mcx + (c^2- a^2) = 0$
Here,
$a = (1 + m^2), b = 2mc$ and $c = (c^2 - a^2)$
It is given that the roots of the equation are equal; therefore, we have:
$D = 0$
$\Rightarrow (b^2 - 4ac) = 0$
$\Rightarrow (2mc)^2 - 4 \times (1 + m^2) \times (c^2 - a^2) = 0$
$\Rightarrow 4m^2c^2 - 4(c^2 - a^2 + m^2c^2- m^2a^2) = 0$
$\Rightarrow 4m^2c^2 - 4c^2 + 4a^2 - 4m^2c^2 + 4m^2a^2 = 0$
$\Rightarrow -4c^2 + 4a^2 + 4m^2a^2 = 0$
$\Rightarrow a^2 + m^2a^2 = c^2$
$\Rightarrow a^2(1 + m^2) = c^2$
$\Rightarrow c^2 = a^2(1 + m^2)$
Hence proved.

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