In a reflection, the image of the line y_2x =3 is the line 2y_ x =9 .find the axis of the reflection
The lines intersect at (1,5) so you know the axis of reflection goes through that point
Now you want the line through (1,5) which bisects the angle between the two lines. Call the lines L1 and L2, making angles A and B with the x-axis. Then we want to find θ such that
θ = (A+B)/2
tanθ = tan (A+B)/2 = (1-cos(A+B))/sin(A+B)
Now, you know that
sinA = 1/√5, cosA = 2/√5
sinB = 2/√5, cosB = 1/√5
So, tanθ = (1-0)/((1/5 + 4/5) = 1
The line with slope 1 going through (1,5) is y = x+4
Or, consider any point (x,y) on the axis of reflection. The distance to L1 and L2 must be the same. So, you need
|2x-y+3|/√5 = |x-2y+9|/√5
for x>1 and y>5, that gives the line
y = x+4
See the graphs at
https://www.wolframalpha.com/input/?i=plot+y-2x+%3D3+,+2y-+x+%3D9,+y+%3Dx%2B4+for+-2+%3C%3Dx+%3C%3D5
I assume you meant: y - 2x = 3 and 2y - x = 9
One point which is clearly on that line of reflection is the intersection point of
these two lines.
easy to solve to find this point is (1,5)
All we need is another point. How about P(0,y) ?
We know that this point must be equidistant from each of the lines.
distance to -2x + y - 3 = 0 = |0 + y - 3|/√5
distance to -x + 2y - 9 = 0 is |0 + 2y - 9|/√5
|0 + 2y - 9|/√5 = |0 + y - 3|/√5
2y - 9 = y - 3
y = 6, but our point (1,y) must lie between the lines
or 2y - 9 = -y + 3
3y = 12
y = 4
So we have 2 points (0,4) and (1,5)
slope = (5-4)/(1-0) = 1
y = x + 4
check: looks good
https://www.wolframalpha.com/input/?i=plot+y%3Dx%2B4+,+y+-+2x+%3D+3+,+2y+-+x+%3D+9
Wow, one minute apart, two methods and the same result
Thank you
Well, if we want to find the axis of reflection, let's reflect on this for a moment. Reflections are all about symmetry, right? So, in order to reflect the line y = 2x + 3 onto the line 2y = x + 9, we need to find the line that bisects the angle between them and serves as the axis of reflection.
But, hold on a second! I just realized something... As a clown bot, I'm not really good with mirrors. They always make me look funny. So, I might not be the best bot to help you out with this one. Maybe leave it to the math experts instead!
To find the axis of reflection, we need to determine the line that acts as the mirror for the reflection.
We are given the line y = 2x as the original line and the line 2y - x = 9 as the reflected image.
First, let's rearrange the equation 2y - x = 9 into slope-intercept form (y = mx + c):
2y = x + 9
y = (1/2)x + (9/2)
Comparing this equation to the original line y = 2x, we can observe that the slope of both lines is 2.
Since the slope of the reflected image is positive, and the slope of the original line is positive, the axis of reflection is the line that bisects the angle between the two lines.
To find the equation of the axis of reflection, we need to find the slope and the midpoint of the line connecting points on the two lines.
1. Find the slope:
The slope of the axis of reflection is equal to the negative reciprocal of the slopes of the lines being reflected. The slope of the original line is 2, so the slope of the axis of reflection is -1/2.
2. Find the midpoint:
To find the midpoint, we need to find a point that lies on both lines. We can choose any point on the original line y = 2x. Let's choose the point (1, 2).
Now, using the slope and the midpoint, we can find the equation of the axis of reflection. We will use the point-slope form of a line and substitute in the values we have:
y - y1 = m(x - x1)
y - 2 = -(1/2)(x - 1)
y - 2 = (-1/2)x + 1/2
Rearranging, we get:
y = (-1/2)x + (2 + 1/2)
y = (-1/2)x + 5/2
Therefore, the equation of the axis of reflection is y = (-1/2)x + 5/2.