To find the equation of the bisector of the acute angle between the lines 2x + y = 4 and 4y + 2x = 5, you can follow these steps:
1. Find the slopes (m1 and m2) of both lines.
The lines are given in the form Ax + By = C, where the slope is -A/B. So, for the first line:
m1 = -2/1 = -2
For the second line:
m2 = -2/4 = -1/2
2. Find the average of the slopes (m1 and m2):
m = (m1 + m2) / 2 = (-2 - 1/2) / 2 = (-4 - 1) / 4 = -5/4
3. Find the perpendicular bisector of the lines using the average slope (m) and the midpoint of the segment connecting the two lines.
To find the midpoint, first, find the intersection point of the two lines:
Solve the system of equations:
2x + y = 4
4y + 2x = 5
This yields x = 3/4 and y = 5/4, so the midpoint is (3/4, 5/4).
4. Now, we have the midpoint (x₀, y₀) = (3/4, 5/4) and the slope (m) = -5/4. We can use the point-slope form of a line to find the equation of the bisector:
y - y₀ = m(x - x₀)
Plugging in the values:
y - 5/4 = (-5/4)(x - 3/4)
Now, you can simplify and rearrange the equation to a more standard form:
4y - 5 = -5(x - 3/4)
4y - 5 = -5x + 15/4
4y = -5x + 15/4 + 5
4y = -5x + 35/4
Multiply both sides by 4 to eliminate fractions:
16y = -20x + 35
Now, you have the equation of the bisector of the acute angle between the given lines: 16y = -20x + 35.