Question

If the lines $x + 3y - 9 = 0,\,{\text{4x + by - 2 = 0}}$ and $2x - y - 4 = 0$ are concurrent, then b is equal to
$
  1) - 5 \\
  2)5 \\
  3)1 \\
  4)0 \\
$

Answer 1

Hint: Here we are given that the three lines are concurrent and so it suggests that the determinant is equal to zero. Take a determinant equal to zero for the three lines and then expand the determinant and simplify the equation for the resultant required value for “b”.

Complete step-by-step answer:
Given three lines are equal to concurrent so the determinant is equal to zero.
$\left| {\begin{array}{*{20}{c}}
  1&3&{ - 9} \\
  4&b&{ - 2} \\
  2&{ - 1}&{ - 4}
\end{array}} \right| = 0$
Expand the above determinant –
$ \Rightarrow 1( - 4b - 2) - 3( - 16 + 4) - 9( - 4 - 2b) = 0$
Simplify the above expression. When you combine two terms with the negative sign then you have to add the terms to give a negative sign to the resultant value. When you combine one negative and one positive term, you have to do subtraction and give the sign of the bigger term to the resultant value.
$ \Rightarrow - 4b - 2 - 3( - 14) - 9( - 4 - 2b) = 0$
Product of two negative terms gives the positive term.
$ \Rightarrow - 4b - 2 + 36 + 36 + 18b = 0$
Make the term with “b” on the left hand side and the constants on the right hand side of the equation. When you move any term from one side to the opposite side then the sign of the terms changes. Positive term becomes negative and vice-versa.
$ \Rightarrow - 4b + 18b = 2 - 36 - 36$
Simplify the above expression combining the like terms on both the sides of the equation –
$ \Rightarrow 14b = - 70$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow b = - \dfrac{{70}}{{14}}$
Remove common factors from the numerator and the denominator.
$ \Rightarrow b = - 5$
From the given multiple choices – the first option is the correct answer.
So, the correct answer is “Option 1”.

Note: Be careful about the sign convention while simplification. When there is a negative sign outside the bracket then the sign of the terms inside the bracket changes when opened. Positive term becomes negative and vice-versa. When there is a positive sign outside the bracket then the sign of the terms remains the same.