Answer
Verified
424.2k+ views
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.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE