Straight Lines
Concurrency and angle bisectors
Grade 11
Question:
<p>Three lines \(2x + 11y - 5 = 0\), \(24x + 7y - 20 = 0\) and \(4x - 3y - 2 = 0\) are given. Which of the following is correct?</p>
<p>They form a triangle</p>
<p>They are concurrent and one of them bisects the angle between the other two</p>
<p>They are parallel</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Find the intersection point of two lines, then verify if it lies on the third line. Three lines are concurrent if they pass through a common point.
<p><strong>Step 1:</strong> Find intersection of lines (1) and (2):</p><p>2x + 11y - 5 = 0 ... (1)</p><p>24x + 7y - 20 = 0 ... (2)</p><p>Multiply (1) by 12: 24x + 132y - 60 = 0</p><p>Subtract (2): 125y - 40 = 0 → y = 8/25</p><p>Substitute in (1): 2x + 11(8/25) - 5 = 0 → 2x = 5 - 88/25 = 37/25 → x = 37/50</p><p><strong>Step 2:</strong> Check if point (37/50, 8/25) satisfies line (3): 4x - 3y - 2 = 0</p><p>4(37/50) - 3(8/25) - 2 = 148/50 - 24/25 - 2 = 148/50 - 48/50 - 100/50 = 0 ✓</p><p><strong>Step 3:</strong> Since the point of intersection of lines (1) and (2) lies on line (3), all three lines are concurrent.</p><p>∴ Answer: B (The three lines are concurrent)</p>
Correct Answer: B