To find the equation of the line that minimizes the area of the quadrilateral formed by points A, A', B, and B', we need to analyze the geometry of the situation involving the two parallel lines and the perpendiculars drawn from points A and B. Let's break this down step by step.
Understanding the Lines
The lines given are:
- L1: 3x + y = 5
- L2: 3x + y = 15
Both lines have the same slope, which is -3, indicating they are parallel. The distance between these two lines can be calculated, but for our purpose, we will focus on the area formed by the perpendiculars dropped from points A and B.
Finding the Intersection Points
Let’s denote the line we want to find as line L. This line passes through the point (4, 3) and intersects L1 at point A and L2 at point B. The general equation of line L can be expressed in slope-intercept form as:
y - 3 = m(x - 4),
where m is the slope of line L.
Finding Points A and B
To find points A and B, we need to substitute the equation of line L into the equations of L1 and L2. For L1:
Substituting into L1:
3x + (m(x - 4) + 3) = 5.
Solving this will give us the coordinates of point A. Similarly, we can find point B by substituting into L2:
3x + (m(x - 4) + 3) = 15.
Perpendiculars from A and B
Once we have points A and B, we can find the equations of the perpendicular lines dropped from these points to L1 and L2. The slope of the perpendicular line will be the negative reciprocal of the slope of L1 and L2, which is 1/3.
Using the point-slope form, we can find the equations of the lines from A and B to their respective perpendicular intersections A' and B'.
Area of Quadrilateral AA'BB'
The area of the quadrilateral formed by points A, A', B, and B' can be calculated using the formula for the area of a trapezoid or by using the coordinates of the vertices. The goal is to minimize this area with respect to the slope m of line L.
Minimizing the Area
To minimize the area, we can set up a function that represents the area in terms of m and then take the derivative of this function. Setting the derivative equal to zero will give us the value of m that minimizes the area.
After performing the calculations, we find that the optimal slope corresponds to the equation:
7x + y - 31 = 0.
Conclusion
This equation represents the line that minimizes the area of the quadrilateral AA'BB'. The reasoning behind this involves understanding the geometry of the situation, the relationships between the lines, and the optimization of the area formed by the intersections and perpendiculars. By carefully analyzing the slopes and distances, we arrive at the desired equation.
