Question

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

Answer 1

The axis of reflection is the perpendicular bisector of the segment connecting any point on the original line to its corresponding point on the reflected line.

To find a convenient point on the original line, let's first rewrite the equation of the line y-2x=3 in slope-intercept form:
y = 2x + 3

One point on this line is (0, 3). Its corresponding point on the reflected line is the intersection of the two lines. We can find this point by solving the system of equations:

y - 2x = 3
2y - x = 9

Multiplying the first equation by 2 and adding it to the second equation, we get:

5y = 15

So y = 3. Substituting into the first equation, we get:

3 - 2x = 3

So x = 0.

Therefore, the corresponding point is (0, 3).

Now we can find the midpoint of the segment connecting (0, 3) to its reflection across the axis of reflection (let's call it (a, b)). The midpoint of this segment lies on the axis of reflection.

The midpoint is:

((0 + a)/2, (3 + b)/2)

And its reflection across the axis of reflection is:

((a + 2b - 18)/5, (2a + b - 3)/5)

Setting these two points equal (since they are reflections across the axis of reflection), we get the system:

(a + 2b - 18)/5 = (0 + a)/2
(2a + b - 3)/5 = (3 + b)/2

Solving this system, we get:

a = 12
b = 1

Therefore, the axis of reflection is the line passing through the point (12, 1) and perpendicular to the original line y-2x=3.

The original line has slope 2, so the axis of reflection has slope -1/2 (since it is perpendicular).

Using point-slope form with the point (12, 1), we get the equation of the axis of reflection:

y - 1 = (-1/2)(x - 12)

Simplifying, we get:

2y - x = 14

Therefore, the axis of reflection is the line 2y - x = 14.