In a reflection the image of the line y-2x=3 is the line 2y-x=9. Find the axis of reflection.
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.
To find the axis of reflection, we need to understand how reflection works and how it affects the equation of lines.
When a line is reflected across another line, the image of the line is the same as the original line but on the other side of the reflection axis. Additionally, the reflection axis is the perpendicular bisector of any line segment connecting a point on the original line to its corresponding image.
In this case, we need to understand the relationship between the given equations of the lines: y - 2x = 3 and 2y - x = 9.
First, let's transform the equation of the line y - 2x = 3 into slope-intercept form (y = mx + b) by solving for y:
y - 2x = 3
y = 2x + 3
Now, let's analyze the equation 2y - x = 9:
2y - x = 9
2y = x + 9
y = (x + 9)/2
Comparing the two equations, we can see that they have the same slope but different y-intercepts. This tells us that the lines are parallel. In other words, the image of the line y - 2x = 3 is parallel to the line 2y - x = 9.
Since the image of the line and the original line are parallel, the axis of reflection must be a line with a slope perpendicular to the slopes of these lines.
The slope of the original line is 2, so the slope of the axis of reflection would be -1/2 (the negative reciprocal of 2).
Therefore, the axis of reflection is a line with a slope of -1/2.