Straight Lines
Angle Between Two Lines
Grade 11
Question:
<p>In triangle ABC, the angle between lines AC and AB can be found using the formula for the angle between two lines. If the slope of AC is \(\frac{1}{4}\) and tan θ₁ = \(\frac{1}{3}\), find the slope m of AB such that tan θ₂ = \(\frac{1}{m}\).</p>
Step-by-Step Solution
Key Concept: The angle between two lines with slopes m₁ and m₂ is given by tan θ = |(m₁ - m₂)/(1 + m₁m₂)|. Consider both cases based on the relative magnitudes.
<p><strong>Step 1:</strong> Given: tan θ₁ = \(\frac{1}{3}\) (angle of AC) and tan θ₂ = \(\frac{1}{m}\) (angle of AB)</p><p><strong>Step 2:</strong> Formula for angle between two lines: \(\tan(\theta_1 - \theta_2) = \frac{\tan\theta_1 - \tan\theta_2}{1 + \tan\theta_1\tan\theta_2}\)</p><p><strong>Case-I:</strong> If m < 3:</p><p>\(\frac{1}{3} = \frac{\frac{1}{3} - \frac{1}{m}}{1 + \frac{1}{3m}}\)</p><p>Solving: \(m = \frac{3}{4}\)</p><p><strong>Case-II:</strong> If m > 3:</p><p>\(\frac{3}{m} = \frac{1}{4}\)</p><p>Solving: \(m = 12\)</p>
Correct Answer: m = 12 or m = 3/4