Question

HM between the roots of the equation \[{x^2} - 10x + 11 = 0\] is
A. \[\dfrac{1}{5}\]
B. \[\dfrac{5}{{21}}\]
C. \[\dfrac{{21}}{{20}}\]
D. \[\dfrac{{11}}{5}\]

Answer 1

Hint: Here we are asked to find the harmonic mean of the roots of the given equation. The harmonic mean of the given values is nothing but the reciprocal of the arithmetic mean of the given values. First, we will find the roots of the given equation since the given equation is a quadratic equation it will have two roots. Then we will find the harmonic equation of those roots using the formula.

Formula Used: If \[\alpha \]and \[\beta \] are the roots then the quadratic equation can be written as\[{x^2} - (\alpha + \beta )x + \alpha \beta = 0\].
The harmonic mean of \[a\]and \[b\]\[ = \dfrac{{2(ab)}}{{a + b}}\]

Complete step-by-step solution:
The given quadratic equation is \[{x^2} - 10x + 11 = 0\] we aim to find the harmonic mean of this equation. For that, we need to first find the roots of this equation. We know that if \[\alpha \]and \[\beta \]are the roots then the quadratic equation can be written as \[{x^2} - (\alpha + \beta )x + \alpha \beta = 0\] . On comparing the given equation with the general equation, we get the sum of the roots \[\alpha + \beta = 10\]and the product of the root\[\alpha \beta = 11\].
And we know that the formula for harmonic mean is \[\dfrac{{2(ab)}}{{a + b}}\], here we need to find the HM of the roots of the given equation that is \[\alpha \]and \[\beta \].
Thus, HM of \[\alpha \]and \[\beta \]\[ = \dfrac{{2(\alpha \beta )}}{{\alpha + \beta }}\]
We already know the values of \[\alpha \beta \] and\[\alpha + \beta \] . Substituting these in the formula we get
HM of \[\alpha \]and \[\beta \] \[ = \dfrac{{2(11)}}{{10}}\]
On simplifying the above, we get
HM of \[\alpha \]and \[ = \dfrac{{11}}{5}\]
Thus, we have found the value of the harmonic mean of the roots of the given equation. Now let’s see the options to find the correct answer.
Option (a) \[\dfrac{1}{5}\] is an incorrect option as we got the HM as \[\dfrac{{11}}{5}\] in or calculation.
Option (b) \[\dfrac{5}{{21}}\] is an incorrect option as we got the HM as \[\dfrac{{11}}{5}\] in or calculation.
Option (c) \[\dfrac{{21}}{{20}}\] is an incorrect option as we got the HM as \[\dfrac{{11}}{5}\] in or calculation.
Option (d) \[\dfrac{{11}}{5}\] is the correct option as we got the same value in our calculation.
Hence, option (d) \[\dfrac{{11}}{5}\] is the correct answer.

Note:We can also find the sum of roots and product of roots of a quadratic equation using the following method: If \[a{x^2} + bx + c = 0\] is a quadratic equation then the sum of its roots \[ = \dfrac{{ - b}}{a}\] and product of its root \[ = \dfrac{c}{a}\].