Question

In the following equation, find the value of $\dfrac{{\text{k}}}{{{\text{ab}}}}$for which the given value is a solution of the given equation: ${{\text{x}}^2} - {\text{x}}\left( {{\text{a + b}}} \right) + {\text{k = 0}}$, given value x = a.

Answer 1

Hint: In order to compute$\dfrac{{\text{k}}}{{{\text{ab}}}}$, we try to find the solution of the given equation using the given value of x, i.e. x = a, using the result obtained we find the required value.

Complete Step-by-Step solution:
Given data, ${{\text{x}}^2} - {\text{x}}\left( {{\text{a + b}}} \right) + {\text{k = 0}}$ and x = a.
We substitute x = a in the given equation, we get
$
   \Rightarrow {{\text{a}}^2} - {\text{a}}\left( {{\text{a + b}}} \right) + {\text{k = 0}} \\
   \Rightarrow {{\text{a}}^2} - {{\text{a}}^2} - {\text{ab + k = 0}} \\
   \Rightarrow {\text{k = ab}} \\
$
We are supposed to find the value of $\dfrac{{\text{k}}}{{{\text{ab}}}}$
$ \Rightarrow \dfrac{{\text{k}}}{{{\text{ab}}}} = \dfrac{{{\text{ab}}}}{{{\text{ab}}}} = 1$
Hence the value of $\dfrac{{\text{k}}}{{{\text{ab}}}}$= 1.

Note: In order to solve questions of this type the key is to understand that for a value to be the solution of the equation, it definitely has to satisfy the equation. Given x = a and x is the solution of the equation, which is why we substituted it and found the value of k. That line is the key to solve this problem. After we found k we directly substitute it to determine the answer.