Question

Find the slope of the line perpendicular to AB where A (5, -6) and B (2, -7).
A. -2
B. -3
C. 1
D. 2

Answer 1

Hint: We will suppose the slope of the line AB and its perpendicular line to be variable. Now we will use the formula to find the value of slope of any line passing through two points, say \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] then slope is given by:
Slope \[=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\]
Then we will use the property that that product of the slope of a line and the line perpendicular to it is equal to (-1).

Complete step-by-step answer:

We have been given two points A (5, -6) and B (2, -7).
Since we know the slope of a line passing through two points say \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are given as:
Slope \[=\left( \dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \right)\]
So slope of AB line \[=\left( \dfrac{-7-(-6)}{2-5} \right)=\dfrac{-7+6}{-3}=\dfrac{-1}{-3}=\dfrac{1}{3}\]
Now we have been asked to find the slope of a line which is perpendicular to the line AB.
Let the slope of the line AB be \[{{m}_{1}}\] and slope of line perpendicular to it be \[{{m}_{2}}\].
We have \[{{m}_{1}}=\dfrac{1}{3},{{m}_{2}}=?\]
Since we know that the product of slopes of a line and a line perpendicular to it is equal to (-1)
\[\begin{align}
  & \Rightarrow {{m}_{1}}{{m}_{2}}=-1 \\
 & \Rightarrow \dfrac{1}{3}\times {{m}_{2}}=-1 \\
 & \Rightarrow {{m}_{2}}=-3 \\
\end{align}\]
Hence the slope of the line perpendicular to AB is equal to -3.
Therefore, the correct option of the question is option B.

Note: Be careful while calculating the slope of a line AB and also take care of the sign as there is a chance of a sign mistake. Also, remember the property that the product of slopes of a line and a line perpendicular to it is equal to (-1).