The intersection point of the given lines is (0, 2). The bisector's equation is y - 2 = (√3/3)x. However, there appears to be an issue, as points p1 and p2 cannot be determined at a distance of 5 units from the intersection point.
The given equation is y - √3|x| = 2. To find the points of intersection, we'll consider the two cases separately for x ≥ 0 and x < 0:
For x ≥ 0, the equation becomes y - √3x - 2 = 0. Solving this equation, we get y = √3x + 2.
For x < 0, the equation becomes y + √3x - 2 = 0. Solving this equation, we get y = -√3x + 2.
Now, let's find the intersection point by equating y in both equations:
√3x + 2 = -√3x + 2
√3x = -√3x
x = 0
Now, substitute x = 0 back into either equation to find y:
y = 2
So, the point of intersection is (0, 2).
Next, we need the equation of the bisector. The given line can be written as y = √3|x| + 2, which has a slope of √3. The bisector's slope can be found using the formula tan(θ/2) = (m1 - m2) / (1 + m1m2), where m1 = √3 and m2 is the slope of the bisector.
Solving for m2:
tan(θ/2) = (√3 - m2) / (1 + √3m2)
The solution for m2 is found to be m2 = 1/√3 = √3/3.
Now, we can use the point-slope form to find the equation of the bisector. Let (x1, y1) = (0, 2) be the point of intersection:
y - y1 = m(x - x1)
y - 2 = (√3/3)x
Now, let's find the points p1 and p2 at a distance of 5 units from the intersection point (0, 2). The distance formula is given by d = √((x2 - x1)^2 + (y2 - y1)^2).
Let p1 have coordinates (x1, y1) and p2 have coordinates (x2, y2). The distance between p1 and the intersection point is 5 units:
√((x1 - 0)^2 + (y1 - 2)^2) = 5
Substitute x1 = 0 and y1 = 2 into the equation:
√(0 + (2 - 2)^2) = 5
0 = 5 (This is not possible)
Therefore, there is an issue, and it seems there is no solution for points p1 and p2 at a distance of 5 units from the intersection point.