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Q. 126

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Found in: Page 43

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# 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.

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

See the step by step solution

## Step 1: Given information

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

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

We get

Slope$=\frac{a-b}{b-a}\phantom{\rule{0ex}{0ex}}=-1$

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 $\left(a,b\right)\left(b,a\right)$is perpendicular to line $y=x$

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

We get,

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

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 $y=x$