In a reflection the image of the line y_2x=3 is the line 2y_x=9. Find the axis of reflection.
First, let's rewrite the equations of the two lines in slope-intercept form:
y = (1/2)x + 3/2 (for y_2x = 3)
y = (4/2)x + 9/2 = 2x + 9/2 (for 2y_x = 9)
To find the axis of reflection, we need to find the equation of the line that is perpendicular to y_2x = 3 and passes through its midpoint. This line will be the axis of reflection.
The slope of y_2x = 3 is -2, so the slope of the perpendicular line is 1/2. The midpoint of y_2x = 3 is at (0, 3/2). Using the slope-intercept form of a line, the equation of the perpendicular line is:
y - 3/2 = 1/2(x - 0)
Simplifying, we get:
y = (1/2)x + 3/2
This is the equation of the axis of reflection. We can confirm that the line y_2x = 3 is reflected across this line to become 2y_x = 9 by checking that the distance between each point on the original line and the axis of reflection is equal to the distance between its corresponding point on the reflected line and the axis of reflection.
To find the axis of reflection, we need to find the equation of the reflection line.
Given that the image of the line y = 2x + 3 is the line 2y = x + 9, we can see that the reflection line is the perpendicular bisector of the line segment connecting these two lines.
The equation of the line y = 2x + 3 can be rewritten in slope-intercept form as y = (2/1)x + 3.
The slope of this line is 2, and any line perpendicular to it will have a slope equal to the negative reciprocal of 2, which is -1/2.
Using the slope-intercept form, the equation of the perpendicular line can be written as y = (-1/2)x + b, where b is the y-intercept of the line.
Since the perpendicular line is the reflection line, it will pass through the midpoint of the line segment connecting the two given lines.
To find the midpoint, we need to find the intersection point of the two given lines:
2y = x + 9 and y = 2x + 3.
Solving these equations simultaneously, we get:
2(2x + 3) = x + 9
4x + 6 = x + 9
3x = 3
x = 1
Substituting the value of x back into one of the given equations, we find:
y = 2(1) + 3
y = 5
So the midpoint of the line segment connecting the two given lines is (1, 5).
Now, substituting these coordinates into the equation y = (-1/2)x + b, we can solve for b:
5 = (-1/2)(1) + b
5 = -1/2 + b
b = 5 + 1/2
b = 11/2
Therefore, the equation of the reflection line, or the axis of reflection, is y = (-1/2)x + 11/2.