in a reflection the image of the line y_2x=3 is the line 2y_x=9 find the axis of reflection
The axis of reflection will be midway between the two lines, right?
These two lines intersect at (6,15)
The axis must have y-intercept at (0,6)
So, now just find the line through those two points.
To find the axis of reflection for a given reflection, we need to identify the line that acts as the mirror or reflection axis.
In this case, we have two lines:
1. The original line: y = 2x + 3
2. The image line after reflection: 2y = x + 9
Step 1: Convert both equations to slope-intercept form (y = mx + b):
1.1. Original line: y = 2x + 3
1.2. Image line: 2y = x + 9 --> Divide both sides by 2: y = 0.5x + 4.5
Step 2: Compare the slopes of the original and image lines:
The original line has a slope of 2, and the image line has a slope of 0.5.
Step 3: Determine the axis of reflection:
The axis of reflection is perpendicular to both lines and bisects the angle between them. In other words, it is a line that is perpendicular to both lines.
To find the axis of reflection, we need to find the equation of a line that is perpendicular to both lines. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.
The negative reciprocal of 2 is -1/2, and the negative reciprocal of 0.5 is -2.
Therefore, the axis of reflection has a slope of -1/2.
Step 4: Find the y-intercept of the axis of reflection:
To find the y-intercept, we need a point that lies on the axis of reflection. One option is to find the intersection point between the two lines.
Solving the equations y = 2x + 3 and y = -1/2x + b, we get:
2x + 3 = -1/2x + b
2.5x = b - 3
x = (b - 3) / 2.5
Since this point lies on the axis of reflection, the y-coordinate should be the same as the x-coordinate:
x = (b - 3) / 2.5
x = (1/2)x - 3/2.5
5x = x - 6
4x = -6
x = -6/4
x = -3/2
Therefore, the point (-3/2, -3/2) lies on the axis of reflection.
Step 5: Find the equation of the axis of reflection:
Using the point-slope form of a line, we can express the equation of the axis of reflection as follows:
y - y₁ = m(x - x₁)
y - (-3/2) = -1/2(x - (-3/2))
y + 3/2 = -1/2(x + 3/2)
2y + 3 = -x - 3/2
2y = -x - 9/2
x + 2y + 9/2 = 0
Thus, the equation x + 2y + 9/2 = 0 represents the axis of reflection for the given reflection.