$(1+m^2)x^2 + 2cmx + (c^2 \,–\, a^2) = 0$ is a quadratic equation and the roots of this equation are same.

01

Due to the equal roots of the quadratic equation, the discriminant of the quadratic equation is zero. For example, $ax^2+bx+c = 0$ is a quadratic equation in general form. The discriminant is $b^2-4ac$ and it is zero due to equal roots.

$b^2 \,-\, 4ac = 0$

02

Compare the quadratic equation $(1+m^2)x^2 + 2cmx + (c^2 \,–\, a^2) = 0$ with standard form quadratic equation $ax^2+bx+c = 0$.

$a = 1+m^2$, $b = 2cm$ and $c = (c^2 \,–\, a^2)$. Substitute these values in discriminate of the quadratic equation.

$b^2 \,-\, 4ac = 0$

$\implies {(2cm)}^2 \,-\, 4 \times (1+m^2) \times (c^2 \,–\, a^2) = 0$

$\implies 4c^2m^2 \,-\, 4 \times (c^2 \,–\, a^2 + m^2 c^2 \,–\, m^2 a^2) = 0$

$\implies 4c^2m^2 \,-\, 4c^2 + 4a^2 \,-\, 4m^2c^2 + 4m^2a^2 = 0$

$\implies 4c^2m^2 \,-\, 4c^2 + 4a^2 \,-\, 4c^2m^2 + 4m^2a^2 = 0$

$\require{cancel} \implies \cancel{4c^2m^2} \,-\, 4c^2 + 4a^2 \,-\, \cancel{4c^2m^2} + 4m^2a^2 = 0$

$\implies \,-\, 4c^2 + 4a^2 + 4m^2a^2 = 0$

$\implies 4a^2 + 4m^2a^2 = 4c^2$

$\implies 4c^2 = 4a^2 + 4m^2a^2$

$\implies 4c^2 = 4a^2(1 + m^2)$

$\require{cancel} \implies \cancel{4}c^2 = \cancel{4}a^2(1 + m^2)$

$\implies c^2 = a^2(1 + m^2)$

$\therefore \,\,\,\,\,\, c^2 -a^2(1 + m^2) = 0$

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