Question

The line passing through the points $\left( 3,-4 \right)$ and $\left( -2,6 \right)$ and a line passing through $\left( -3,6 \right)$ and $\left( 9,-18 \right)$ are
A. Perpendicular
B. parallel
C. makes an angle ${{60}^{\circ }}$ with each other
D. None of these

Answer 1

Hint: We first use the condition that equal slopes of two straight lines make them parallel. We use the condition of $m=\dfrac{d-b}{c-a}$ for the slopes and find the relation between the lines. Equal slopes give us the final solution.

Complete step by step answer:
We know that if the slopes of two straight lines are equal then the lines are parallel.We also know that the slope of a line passing through points $\left( a,b \right)$ and $\left( c,d \right)$ will be $m=\dfrac{d-b}{c-a}$.Therefore, slope of a line passing through points $\left( 3,-4 \right)$ and $\left( -2,6 \right)$ will be
${{m}_{1}}=\dfrac{6-\left( -4 \right)}{-2-3} \\
\Rightarrow {{m}_{1}}=\dfrac{10}{-5} \\
\Rightarrow {{m}_{1}}=-2$
and slope of a line passing through points $\left( -3,6 \right)$ and $\left( 9,-18 \right)$ will be
\[{{m}_{2}}=\dfrac{\left( -18 \right)-6}{9-\left( -3 \right)} \\
\Rightarrow {{m}_{2}}=\dfrac{-24}{12} \\
\therefore {{m}_{2}}=-2\]
Since slope for both the lines are equal and therefore the lines are parallel.

Hence, the correct option is B.

Note: A line parallel to the X-axis does not intersect the X-axis at any finite distance and hence we cannot get any finite x-intercept of such a line. Same goes for lines parallel to the Y-axis. In case of slope of a line the range of the slope is 0 to $\infty $.