In a reflection, the image of the line y-2x=3 is the line 2y-x=9.Find the axis of reflection.
Look at the original lines first.
https://www.wolframalpha.com/input/?i=graph+y-2x%3D3%2C+2y-x%3D9
As you can see, there would be two such lines, one will have a positive slope, the other will have a negative slope.
To be on the axis of symmetry, any point (X,Y) must be equidistant from each of the original lines.
Rewriting the origianls as 2x - y + 3 =0 and x - 2y + 9 = 0, and using the "distance from a point to a line" formula, we get
|2X - Y + 3|/√5 = |X - 2Y + 9|/√5
giving us X + Y - 6 = 0 and X - Y + 4 = 0
In terms of the standard (x,y) we have : x + y = 6 and x - y = -4
confirm:
https://www.wolframalpha.com/input/?i=graph+y-2x%3D3%2C+2y-x%3D9%2C+x%2By+%3D+6%2C+x-y+%3D+-4
To find the axis of reflection, we need to determine the line about which the reflection occurs. In this case, we need to find a line that is perpendicular to the given line.
Step 1: Convert the given equation of the line y - 2x = 3 into slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
To convert the equation into slope-intercept form, we isolate y:
y - 2x = 3
y = 2x + 3
So the slope-intercept form of the given line is y = 2x + 3, where the slope (m) is 2.
Step 2: Determine the slope of the line perpendicular to the given line.
The slope of the line perpendicular to another line is the negative reciprocal of the slope of the given line.
The negative reciprocal of 2 is -1/2.
Step 3: Use the slope-intercept form to find the equation of the line perpendicular to the given line.
Using the slope-intercept form, we know that the equation of a line is y = mx + b, where m is the slope and b is the y-intercept.
We substitute the slope -1/2 into the equation and find the y-intercept (b) by using any point on the line.
Using the point (9, 0), we substitute the values into the equation:
0 = (-1/2)(9) + b
0 = -9/2 + b
b = 9/2
So the equation of the line perpendicular to y = 2x + 3 is y = -1/2x + 9/2.
Step 4: Find the intersection point of the two lines.
The axis of reflection is the line that passes through the point of intersection of the two lines.
We solve the system of equations formed by the two lines:
y = 2x + 3
y = -1/2x + 9/2
By setting the two equations equal to each other, we can solve for x:
2x + 3 = -1/2x + 9/2
Multiplying both sides by 2 to eliminate fractions:
4x + 6 = -x + 9
5x = 3
x = 3/5
Now, substitute the value of x back into one of the equations to find the value of y:
y = 2(3/5) + 3
y = 6/5 + 3
y = 15/5 + 9/5
y = 24/5
So the intersection point of the two lines is (3/5, 24/5).
Step 5: Determine the axis of reflection.
The axis of reflection is the line that passes through the point of intersection of the two lines. Therefore, the equation of the axis of reflection is x = 3/5.
In conclusion, the axis of reflection for the given line y - 2x = 3, reflected to the line 2y - x = 9, is x = 3/5.