Question

State whether the two lines through (6,3) and (1,1) and through (-2,5) and (2,-5) are parallel, perpendicular or neither.

Answer 1

Hint: Find the slope of the lines using the property that the slope of the line joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$. Use the fact that if the slopes of two lines are equal, then they are parallel to each other and if the product of the slopes of two lines is -1, then the lines are perpendicular. Hence determine whether the lines are parallel or perpendicular or neither.

Complete step-by-step answer:
Finding the slope of the line joining (6,3) and (1,1):
We know that the slope of the line joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$.
Here $x_1=6,x_2=1,y_1=3$ and $y_2=1$
Hence the slope of the line is $m={\displaystyle \frac{1-3}{1-6}}={\displaystyle \frac{-2}{-5}}={\displaystyle \frac{2}{5}}$
Finding the slope of the line joining (-2,5) and (2,-5):
We know that the slope of the line joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$.
Here $x_1=-2,x_2=2,y_1=5$ and $y_2=-5$
Hence the slope of the line is $m={\displaystyle \frac{-5-5}{2-(-2)}}={\displaystyle \frac{-10}{4}}={\displaystyle \frac{-5}{2}}$
Product of slope of the lines $={\displaystyle \frac{2}{5}}\times {\displaystyle \frac{-5}{2}}=-1$
Now since the product of the slopes of the two lines is -1, the lines are perpendicular to each other.
Note: [i] Viewing graphically:

As is evident from the graph $AB\mathrm{\perp }CD$
[ii] Alternative solution:
Let the equation of AB be y=mx+c
Since the line passes through (6,3), we have
$6m+c=3$
Also, since the line passes through (1,1), we have
$m+c=1$
Hence, we have
$6m-m=3-1\Rightarrow m={\displaystyle \frac{2}{5}}$
Hence the slope of AB is ${\displaystyle \frac{2}{5}}$
Let the equation of CD be y = mx+c
Since the line passes through (-2,5), we have
$-2m+c=5$
Also, since the line passes through (2,-5), we have
$2m+c=-5$
Hence, we have
$2m+2m=-5-5\Rightarrow m={\displaystyle \frac{-5}{2}}$
Hence the slope of CD is ${\displaystyle \frac{-5}{2}}$
Hence the lines are perpendicular to each other.