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The image of the point with coordinates (1,1) under the reflection across the line y=mx+b is the point with coordinates \((7,2)\) Find m+b.

 Apr 16, 2022
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Rewrite the equation of the line as: mx - y + b = 0.

We know that the midpoint of the line segment connecting (1, 1) and (7, 2) lies on the line.

Coordinates of midpoint = \(\left(\dfrac{1 + 7}2, \dfrac{1 + 2}2\right) = \left(4, \dfrac32\right)\).

Substituting the coordinates of the midpoint into the equation gives \(4m - \dfrac32 + b = 0\) --- (1).

 

Now, note that the line segment connecting (1, 1) and (7, 2) is perpendicular to the line y = mx + b.

Slope of the line segment connecting (1, 1) and (7, 2) = \(\dfrac{2 -1}{7-1} = \dfrac16\).

Since the product of slopes of perpendicular lines is -1, \(\dfrac16 \cdot m = -1\).

Then \(m = -6\).

 

Substituting m = -6 into (1) gives 

\(4(-6)-\dfrac32 + b = 0\\ b = \dfrac{51}2\)

 

Then m + b = -6 + 51/2 = 39/2

 

Graph: https://www.desmos.com/calculator/mtune9rclo

 Apr 17, 2022

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