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Hint: Find the discriminant of the given quadratic equation using the formula that discriminant of quadratic equation of the form \[a{{x}^{2}}+bx+c=0\] is \[{{b}^{2}}-4ac\] and equate it to zero to prove that given equation has equal roots.

Complete step-by-step answer:

We have the quadratic equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\]. We have to prove that the given quadratic equation has equal roots.

We know that a quadratic equation of the form \[a{{x}^{2}}+bx+c=0\] has equal roots if the value of discriminant of the equation is equal to zero. We know that discriminant of the equation of the form \[a{{x}^{2}}+bx+c=0\] is \[{{b}^{2}}-4ac\].

So, we must have \[{{b}^{2}}-4ac=0\].

Substituting \[a=1+{{m}^{2}},b=2mc,c={{c}^{2}}-{{a}^{2}}\] in the above equation, we have the condition \[{{\left( 2mc \right)}^{2}}-4\left( 1+{{m}^{2}} \right)\left( {{c}^{2}}-{{a}^{2}} \right)=0\] for the equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\] to have equal roots.

Simplifying the above expression, we have \[4{{m}^{2}}{{c}^{2}}-4{{c}^{2}}+4{{a}^{2}}-4{{m}^{2}}{{c}^{2}}+4{{m}^{2}}{{a}^{2}}=0\].

Further simplifying the equation, we get \[4\left( {{a}^{2}}+{{m}^{2}}{{a}^{2}}-{{c}^{2}} \right)=0\].

\[\begin{align}

& \Rightarrow {{a}^{2}}+{{m}^{2}}{{a}^{2}}-{{c}^{2}} \\

& \Rightarrow {{a}^{2}}+{{m}^{2}}{{a}^{2}}={{c}^{2}}={{a}^{2}}\left( 1+{{m}^{2}} \right) \\

\end{align}\]

Hence, we must have \[{{c}^{2}}={{a}^{2}}\left( 1+{{m}^{2}} \right)\] for the equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\] to have equal roots.

Note: A quadratic equation is any polynomial equation in which the highest degree of the variable is two and which can be written in the form \[a{{x}^{2}}+bx+c=0\] where \[a\ne 0\]. The values of \[x\] that satisfy the given quadratic equation are called roots or solutions of the quadratic equation. Discriminant of a polynomial is a quantity that depends on the coefficients in the polynomial and helps to determine various properties of the roots. It is widely used in determining roots of the polynomial. For a quadratic equation, if the value of discriminant is positive, then the quadratic equation has unequal and real roots. If the discriminant is equal to zero, then the roots of the quadratic equation are equal. However, if the value of discriminant is negative, then the quadratic equation has imaginary (complex) roots.

Complete step-by-step answer:

We have the quadratic equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\]. We have to prove that the given quadratic equation has equal roots.

We know that a quadratic equation of the form \[a{{x}^{2}}+bx+c=0\] has equal roots if the value of discriminant of the equation is equal to zero. We know that discriminant of the equation of the form \[a{{x}^{2}}+bx+c=0\] is \[{{b}^{2}}-4ac\].

So, we must have \[{{b}^{2}}-4ac=0\].

Substituting \[a=1+{{m}^{2}},b=2mc,c={{c}^{2}}-{{a}^{2}}\] in the above equation, we have the condition \[{{\left( 2mc \right)}^{2}}-4\left( 1+{{m}^{2}} \right)\left( {{c}^{2}}-{{a}^{2}} \right)=0\] for the equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\] to have equal roots.

Simplifying the above expression, we have \[4{{m}^{2}}{{c}^{2}}-4{{c}^{2}}+4{{a}^{2}}-4{{m}^{2}}{{c}^{2}}+4{{m}^{2}}{{a}^{2}}=0\].

Further simplifying the equation, we get \[4\left( {{a}^{2}}+{{m}^{2}}{{a}^{2}}-{{c}^{2}} \right)=0\].

\[\begin{align}

& \Rightarrow {{a}^{2}}+{{m}^{2}}{{a}^{2}}-{{c}^{2}} \\

& \Rightarrow {{a}^{2}}+{{m}^{2}}{{a}^{2}}={{c}^{2}}={{a}^{2}}\left( 1+{{m}^{2}} \right) \\

\end{align}\]

Hence, we must have \[{{c}^{2}}={{a}^{2}}\left( 1+{{m}^{2}} \right)\] for the equation \[\left( 1+{{m}^{2}} \right){{x}^{2}}+2mcx+{{c}^{2}}-{{a}^{2}}=0\] to have equal roots.

Note: A quadratic equation is any polynomial equation in which the highest degree of the variable is two and which can be written in the form \[a{{x}^{2}}+bx+c=0\] where \[a\ne 0\]. The values of \[x\] that satisfy the given quadratic equation are called roots or solutions of the quadratic equation. Discriminant of a polynomial is a quantity that depends on the coefficients in the polynomial and helps to determine various properties of the roots. It is widely used in determining roots of the polynomial. For a quadratic equation, if the value of discriminant is positive, then the quadratic equation has unequal and real roots. If the discriminant is equal to zero, then the roots of the quadratic equation are equal. However, if the value of discriminant is negative, then the quadratic equation has imaginary (complex) roots.

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