Straight Lines
Locus and Family of Lines
Grade 11
Question:
<p>Consider a variable line 'L' which passes through the point of intersection 'P' of the lines \(3x + 4y - 12 = 0\) and \(x + 2y - 5 = 0\) meeting the coordinate axes at points A and B. Find the locus of the middle point of the segment AB:</p>
<p>(a) \(3x + 4y = 4xy\)</p>
<p>(b) \(3x + 4y = 3xy\)</p>
<p>(c) \(4x + 3y = 4xy\)</p>
<p>(d) \(4x + 3y = 3xy\)</p>
Step-by-Step Solution
Key Concept: First find point P by solving the system of two lines. Then use the intercept form of a variable line through P: \(\frac{x}{a} + \frac{y}{b} = 1\). The midpoint of AB is \((\frac{a}{2}, \frac{b}{2})\). Eliminate a and b using the constraint that the line passes through P.
<p>Not provided in source material.</p>
Correct Answer: A