Understanding the Components of a Variable Straight Line

A variable straight line is a line that is drawn through the point of intersection of two specific straight lines, xa+yb=1 and xb+ya=1, and subsequently meets the coordinate axes at two points, A and B.

This definition highlights key characteristics that make the line "variable," implying it can take on different orientations while adhering to these geometric constraints.

To fully grasp what a variable straight line represents in this context, it's essential to break down its defining properties:

1. Fixed Point of Intersection

The core of the definition lies in the variable line passing through a fixed point. This point is the unique intersection of the two given lines:

• Line 1: xa + yb = 1
• Line 2: xb + ya = 1

How to find this intersection point:
By solving these two linear equations simultaneously, we can find the coordinates (x, y) of their intersection.

1. Subtracting the second equation from the first:
   (xa + yb) - (xb + ya) = 1 - 1
x(a - b) + y(b - a) = 0
x(a - b) - y(a - b) = 0
(x - y)(a - b) = 0
2. Assuming a ≠ b (if a=b, the lines are identical unless a=b=0, in which case they are not lines, or they are parallel and distinct, meaning no intersection point unless they are the same line), this implies x = y.
3. Substitute x = y back into either original equation (e.g., xa + yb = 1):
   xa + ya = 1
y(a + b) = 1
y = 1 / (a + b)
   Since x = y, then x = 1 / (a + b).

Therefore, the fixed point of intersection is (1/(a+b), 1/(a+b)). All "variable straight lines" in this specific scenario must pass through this constant point.

2. Meeting the Coordinate Axes at A and B

The second crucial condition is that the variable straight line intersects the X-axis at point A and the Y-axis at point B.

• If a line has an X-intercept p and a Y-intercept q, its equation in intercept form is typically X/p + Y/q = 1.
• Here, point A would be (p, 0) and point B would be (0, q).

Since the variable straight line passes through the fixed intersection point (1/(a+b), 1/(a+b)) and has intercepts p and q, this means:
(1/(a+b))/p + (1/(a+b))/q = 1
Factoring out 1/(a+b):
(1/(a+b)) * (1/p + 1/q) = 1
Thus, 1/p + 1/q = a+b.

This relationship 1/p + 1/q = a+b is the defining characteristic of this family of variable straight lines. As long as p and q satisfy this condition, and the line passes through the fixed point (1/(a+b), 1/(a+b)), it is considered one of these "variable straight lines."

Why is it "Variable"?

The term "variable" signifies that there isn't just one such line, but an infinite family of lines. While each line in this family must pass through the same fixed point and satisfy the intercept condition (1/p + 1/q = a+b), their slopes and specific intercepts (p and q) can change. For instance, p can vary, and q will adjust accordingly to maintain the sum of their reciprocals as a+b. This variability allows for different line orientations, all constrained by the initial conditions.

This concept is often explored in analytical geometry to demonstrate properties of families of lines passing through a common point or satisfying certain geometric conditions.