Question

The pair of lines represented by the equations $2x + y + 3 = 0$ and $4x + ky + 6 = 0$ will be parallel if the value of k is.

Answer 1

First, write the above given equations in the form of $y = mx + c$ and find the slope of both equations.
Now, for two lines to be parallel to each other, the slopes of both lines must be equal.
Thus, compare both slopes and find k.

Complete step-by-step answer:
It is given that the pair of lines are represented by the equations $2x + y + 3 = 0$ and $4x + ky + 6 = 0$ .
We have to find k such that the above given lines become parallel.
Now, writing the above two equations in the standard form as $y = mx + c$ .
$\therefore y = - 2x-3$ and $y = - \dfrac{4}{k}x - 6$ .
Now, for the lines to be parallel, the slopes of the lines must be equal.
Here, slope of line 1 is -2 and slope of line 2 is $\dfrac{{ - 4}}{k}$ .
$\therefore$ Slope of line 1 = Slope of line 2
$\therefore - 2 = \dfrac{{ - 4}}{k}$
$\therefore k = 2$
Thus, the lines represented by the equations $2x + y + 3 = 0$ and $4x + ky + 6 = 0$ will be parallel if $k = 2$ .

Note: Alternate Method:
It is given that the pair of lines are represented by the equations $2x + y + 3 = 0$ and $4x + ky + 6 = 0$ .
We have to find k such that the above given lines become parallel.
Now, we will compare the equations of lines with $ax + by + c = 0$ .
On comparing, we get ${a_1} = 2,{b_1} = 1,{c_1} = 3,{a_2} = 4,{b_2} = k,{c_2} = 6$ .
Now for lines to be parallel, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ .
Here, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{4} = \dfrac{1}{2},\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{k}$ .
Thus, compare $\dfrac{{{a_1}}}{{{a_2}}},\dfrac{{{b_1}}}{{{b_2}}}$ .
$\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \\\ \therefore \dfrac{1}{2} = \dfrac{1}{k} \\\ \therefore k = 2 \\\$
Thus, the lines represented by the equations $2x + y + 3 = 0$ and $4x + ky + 6 = 0$ will be parallel if $k = 2$ .