Question

The roots of the equation \[a{x^2} + bx + c = 0\] will be in reciprocal if :
A. a = b
B. a = bc
C. c = a
D. c = b

Answer 1

Hint: We had to only use the identity of sum of roots \[\left( {\dfrac{{ - b}}{a}} \right)\] and product of roots \[\left( {\dfrac{c}{a}} \right)\]. And after solving that equation we will get the required condition for reciprocal roots.

Complete step-by-step solution -
As we know that the quadratic equation is the polynomial equation which has the highest degree equal to 2 or we can say that the highest power of the variable will be equal to 2.
So, the number of roots of the quadratic equation should also be 2.
Now as we know that we had to find the condition of reciprocal roots.
So, the one root of the equation \[a{x^2} + bx + c = 0\] will be k.
Now as we know that the other root should be reciprocal of it.
And as we know that if \[\dfrac{p}{q}\] is the reciprocal number in which the numerator is p and the denominator is equal to q. Then the reciprocal of the fraction \[\dfrac{p}{q}\] will be equal to \[\dfrac{q}{p}\] i.e. numerator and denominator are interchanged.
So, the reciprocal of k will be \[\dfrac{1}{k}\].
So, the other root of the quadratic equation \[a{x^2} + bx + c = 0\] will be \[\dfrac{1}{k}\].
Now as we know that if \[\alpha \] and \[\beta \] are the roots of any quadratic equation \[a{x^2} + bx + c = 0\], then the sum of the roots i.e. \[\alpha \]+ \[\beta \] will be equal to \[\dfrac{{ - {\text{Coefficient of }}x}}{{{\text{Coefficient of }}{x^2}}}\] or we can say that \[\alpha + \beta = \dfrac{{ - b}}{a}\].
And the product of the roots \[\alpha \beta \] will be equal to \[\dfrac{{{\text{Constant term}}}}{{{\text{Coefficient of }}{x^2}}}\] or we can say that \[\alpha \beta = \dfrac{c}{a}\].
So, now let us find the product of roots of the given equation \[a{x^2} + bx + c = 0\].
So, \[k \times \dfrac{1}{k} = \dfrac{c}{a}\]
\[1 = \dfrac{c}{a}\]
Now cross-multiplying the above equation. We get,
c = a
So, if the roots of the equation \[a{x^2} + bx + c = 0\] are reciprocal to each other then c = a.
Hence, the correct option will be C.

Note:- Whenever we come up with this type of problem, the best way to find the condition for the given result i.e. here reciprocal we should consider that the result is true (like we assumed that the roots are reciprocal). After that we had to assume one root and the other roots will be the reciprocal of it. So, now we had to use the identity of product of roots which states that if \[\alpha \] and \[\beta \] are the roots of any quadratic equation then the value of \[\alpha \beta \] will be equal to \[\dfrac{{{\text{Constant term in the equation}}}}{{{\text{Coefficient of }}{x^2}{\text{ in the equation}}}}\] and after solving this equation we will get the required condition for the reciprocal roots of quadratic equation.