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

Answer 1

To find the axis of reflection, we need to find the line that is equidistant from the original line and its reflection. This line will be the axis of reflection.

First, let's rewrite the equation of the original line y - 2x = 3 in slope-intercept form:

y = 2x + 3

The slope of this line is 2, so the slope of the perpendicular bisector (the line equidistant to the original line and its reflection) will be -1/2.

Let the equation of the perpendicular bisector be y = (-1/2)x + b.

Now we need to find b. Since this line is equidistant to the original line and its reflection, we can choose any point on the original line and find the distance between that point and the perpendicular bisector. We can then set this distance equal to the distance between the image of that point (i.e. the corresponding point on the reflection) and the perpendicular bisector.

Let's choose a point on the original line, such as (0,3). The distance from this point to the perpendicular bisector is given by the formula:

d = |(-1/2)(0) + b - 3|/sqrt(1 + (-1/2)^2)

Simplifying this expression, we get:

d = |b - 9/2|/sqrt(5)/2

Now we need to find the image of this point across the line 2y - x = 9. To do this, we can substitute y = (x+9)/2 into the equation of the original line:

(x+9)/2 - 2x = 3

Solving for x, we get:

x = -5

Substituting this value of x into the equation of the reflection line, we get:

2y - (-5) = 9

2y = 14

y = 7

So the image of the point (0,3) is (-5,7).

The distance between this point and the perpendicular bisector is:

d' = |(-1/2)(-5) + b - 7|/sqrt(1 + (-1/2)^2)

Simplifying this expression, we get:

d' = |b - 11/2|/sqrt(5)/2

Since the point (0,3) and its image (-5,7) are equidistant to the perpendicular bisector, we have:

d = d'

|b - 9/2|/sqrt(5)/2 = |b - 11/2|/sqrt(5)/2

Simplifying this expression, we get:

|b - 9/2| = |b - 11/2|

This means that either b - 9/2 = b - 11/2 (i.e. b = 10) or b - 9/2 = -(b - 11/2) (i.e. b = 7).

Therefore, the equations of the perpendicular bisector are:

y = (-1/2)x + 10 and y = (-1/2)x + 7.

(Note that these lines are parallel, so they are both equidistant to the original line and its reflection.)

The midpoint of the line segment connecting the two intersection points of the original line and the perpendicular bisector is the axis of reflection.

To find these intersection points, we can solve the system of equations y = 2x + 3 and y = (-1/2)x + 10 (or y = (-1/2)x + 7):

2x + 3 = (-1/2)x + 10 (or 2x + 3 = (-1/2)x + 7)

Solving for x, we get:

x = 7/5 (or x = -4/5)

Substituting this value of x back into either equation, we get:

y = 23/5 (or y = 19/5)

Therefore, the two intersection points are (7/5, 23/5) and (-4/5, 19/5).

The midpoint of this line segment is:

(x1 + x2)/2, (y1 + y2)/2 = (1/5, 21/5)

Therefore, the axis of reflection is the line

x = 1/5.