Find the distance of the lines 4x + 2y + 3 = 0 and 2x + y - 1 = 0.
To find the distance between two lines, we can use the formula derived from the distance between a point and a line.
Step 1: Convert the given equations to slope-intercept form (y = mx + b).
Line 1: 4x + 2y + 3 = 0
- 2y = -4x - 3
y = 2x + 3/2
Line 2: 2x + y - 1 = 0
y = -2x + 1
Step 2: Find the slopes of the lines.
Comparing the equations with y = mx + b, we can see that the slopes (m) of the lines are:
Line 1: m1 = 2
Line 2: m2 = -2
Step 3: Calculate the perpendicular distance between the lines.
The perpendicular distance (d) can be found using the formula:
d = |b1 - b2| / sqrt(1 + m1^2)
where b1 and b2 are the y-intercepts of Line 1 and Line 2, respectively.
b1 = 3/2
b2 = 1
d = |(3/2) - 1| / sqrt(1 + 2^2)
d = |1/2| / sqrt(1 + 4)
d = (1/2) / sqrt(5)
d = (1/2) * (1/sqrt(5))
d = 1 / (2 * sqrt(5))
d = sqrt(5) / (2 * sqrt(5))
d = sqrt(5) / 2
Therefore, the distance between the lines 4x + 2y + 3 = 0 and 2x + y - 1 = 0 is sqrt(5) / 2 units.