Q. 126 - Precalculus Enhanced with Graphing Utilities

Question

Show that the line containing the points (a, b) and (b, a), a  b, is perpendicular to the line y = x. Also show that the midpoint of (a, b) and (b, a) lies on the line y = x.

Step-by-Step Solution

Answer

We show that the lines are perpendicular to line $y=x$ and also we show that the midpoints lie on the line $y=x$ 

Step 1: Given information

We are given that a line contains a point $(a,b),(b,a)$ 

Step 2: We find the slope of line containing points ( a , b ) ( b , a )

We get

Slope$$\begin{gathered} =\frac{a-b}{b-a} \\ =-1 \end{gathered}$$

Therefore the slope is $-1$

Step 3: We find the slope of line y = x and compare them

Comparing the slope with standard equation, we get $slope=1$

And on multiplying both the slopes we get $-1$.

Hence the line containing point $(a,b)(b,a)$is perpendicular to line $y=x$

Step 4: Find the midpoint of the points ( a , b ) ( b , a )

We get,

$M=(\frac{a+b}{2},\frac{b+a}{2})$

And clearly this point lies on $y=x$

Step 5: Conclusion

We proved that the two lines are perpendicular and the midpoint of (a,b) (b,a) lie on the line $y=x$