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)

Answer 1

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Answer 2

Step-by-step explanation:

Given :

The equation,

$(1 + m^2)x^2 + (2mc)x + (c^2 -a^2) = 0$

has equal roots.

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To Prove :

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

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We know that,

for a quadratic equation (equations of the form ax² + bx + c = 0, are termed as quadratic equation,.) which have real & equal roots, will have,

⇒ (Coefficient of x)² - 4(Coefficient of x²)(constant term)        

⇒ $b^2 - 4ac = 0$

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⇒$(2mc)^2 - 4(1+m^2)(c^2 - a^2)=0$

⇒$4m^2c^2 - 4[1(c^2 - a^2) + m^2(c^2 - a^2)]=0$

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

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

⇒$-4c^2+4a^2+4m^2a^2=0$

⇒$4[-c^2+a^2+m^2a^2] = 0$

⇒$-c^2+a^2+m^2a^2 = 0$

⇒$-c^2 + a^2(1 + m^2) = 0$

⇒$a^2(1 + m^2) = c^2$

Hence , Proved.

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