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.
+9519 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.
