Question

What is the perpendicular line to $y=\dfrac{1}{2}x-1$ ?

Answer 1

This type of question depends on the slope-intercept form of a line. The equation of a slope-intercept form of a line is given by $y=mx+c$ where $m$ is the slope and $c$ is the y-intercept. Also, we use the concept of perpendicular lines that is if the two lines are perpendicular to each other, then the product of their slopes is equal to -1. So if two lines are perpendicular to each other with slopes ${{m}_{1}}$ and ${{m}_{2}}$ then ${{m}_{1}}\times {{m}_{2}}=-1$ . Hence, we can find out the value of the slope of another line if the slope of one of the lines is known. Finally by using slope-intercept form once again we can find out the equation of the perpendicular line.

Complete step by step solution:
The given equation of line is $y=\dfrac{1}{2}x-1$ which is in slope-intercept form $y=mx+c$ where $m$ the slope which in this case is $\dfrac{1}{2}$ .
Let’s call this slope ${{m}_{1}}=\dfrac{1}{2}$
For a line to be perpendicular to this line, we must have ${{m}_{1}}\times {{m}_{2}}=-1$ where ${{m}_{2}}$ is the slope of the perpendicular line.

& \Rightarrow {{m}_{2}}=\dfrac{-1}{{{m}_{1}}} \\\ & \Rightarrow {{m}_{2}}=\dfrac{-1}{\left( \dfrac{1}{2} \right)} \\\ & \Rightarrow {{m}_{2}}=(-1)\times 2 \\\ & \Rightarrow {{m}_{2}}=-2 \\\ \end{aligned}$$ Hence, the slope of the perpendicular line is $${{m}_{2}}=-2$$. **So that, the equation of the perpendicular line is given by, $$y=-2x+b$$ where b can be any value.** **Note:** While solving this problem students may make mistakes in calculation of $${{m}_{2}}$$. Instead of using division one may perform multiplication. So when students use basic rules of division and multiplication they have to take care about the change in sign when a number is shifted from left to right or right to left sides.