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In a reflection, the image of the line y-2x=3 is the line 2y-x=9. find the axis of reflection ​

1 Answer

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Answer:

y = x + 4 or y = -x +6

Explanation:

You want the line of reflection that maps y -2x = 3 to 2y -x = 9.

Line of reflection

Points on the line of reflection will be equidistant from both lines. The equation for the distance from a point to a line can be used.

For line ax +by +c = 0, the distance from point (x, y) to that line is ...

d = |ax +by +c|/√(a² +b²)

Then the distances to the lines are the same when ...

|y -2x -3|/√(1² +2²) = |2y -x -9|/√(2² +1²)

Equations

Multiplying by √5 and unfolding the absolute value, we have the two equations ...

  • y -2x -3 = 2y -x -9
  • y -2x -3 = -(2y -x -9)

Simplifying, the first gives ...

x + y = 6

Simplifying the second gives ...

3x -3y = -12

x - y = -4

The equations of the lines of reflection are x+y = 6, or x-y = -4.

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Additional comment

Each of these equations can be written in slope-intercept form, as they are at the top of this answer. They are shown in orange on the attached graph.

Basically, each line bisects the angle formed by the given lines. As you can see, there are two angle bisectors, one for the acute angle, and one for the obtuse angle.

The above solution shows us that general form lines ax+by-c=0 and dx+ey-g=0 will have angle bisectors (lines of reflection) with slopes (a+d)/(-b-e) and (a-d)/(-b+e).

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In a reflection, the image of the line y-2x=3 is the line 2y-x=9. find the axis of-example-1
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