Question

Two lines $x+2y+7=0$ and $2x+ky+18=0$ do not intersect each other, Find the value of
$k$.
(a) $3$
(b) $2$
(c) $1$
(d) $4$

Answer 1

Hint: Two lines are said to be parallel if they never intersect each other or they have equal slopes.

Before proceeding with the question, we must know that any two lines does not intersect
each other if they are parallel to each other. We must know the condition under which the two lines are considered as parallel or two lines never meet each other i.e. does not intersect with each other.
Let us assume any two line on the cartesian plane that are having their equations as,
${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$
The condition under which the above two lines never meet each other or we can say that the
condition under which the above two lines does not intersect with each other is,
$\dfrac{{{a}_{1}}}{{{b}_{1}}}=\dfrac{{{a}_{2}}}{{{b}_{2}}}.........\left( 1 \right)$
In the question, we are given two lines $x+2y+7=0$ and $2x+ky+18=0$. If we compare the
coefficient of $x$ and $y$ of these lines with the coefficient of $x$ and $y$ of the lines which we
assumed above, we can say,
${{a}_{1}}=1$, ${{b}_{1}}=2$, ${{c}_{1}}=7$ and ${{a}_{2}}=2$, ${{b}_{2}}=k$, ${{c}_{2}}=18$
Since we have established a relation between ${{a}_{1}},{{b}_{1}},{{a}_{2}},{{b}_{2}}$ in equation
$\left( 1 \right)$ for the two lines to be parallel, substituting ${{a}_{1}}=1$, ${{b}_{1}}=2$,
${{a}_{2}}=2$, ${{b}_{2}}=k$ in equation $\left( 1 \right)$, we get,
$\dfrac{1}{2}=\dfrac{2}{k}$
$\Rightarrow k=2\times 2$
$\Rightarrow k=4$


Note: There is an alternate method to solve such types of questions. The two lines do not intersect each other if they are parallel to each other. If any two lines are parallel to each other, the slopes of these two lines must be equal. For a line $ax+by+c=0$, the slope is given by $\dfrac{-a}{b}$ . So instead of using equation $\left( 1 \right)$, we can also equate the slopes of the two lines.