Question

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

Answer 1

Hint: 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.
\[\begin{align}
  & \Rightarrow {{m}_{2}}=\dfrac{-1}{{{m}_{1}}} \\
 & \Rightarrow {{m}_{2}}=\dfrac{-1}{\left( \dfrac{1}{2} \right)} \\
 & \Rightarrow {{m}_{2}}=(-1)\times 2 \\
 & \Rightarrow {{m}_{2}}=-2 \\
\end{align}\]
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.