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# Coordinates

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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

Apr 17, 2022