Straight Lines
Angle Between Two Lines
Grade 11
Question:
<p>Find the equations of lines passing through point (2, 3) such that the tangent of the angle between them equals specific trigonometric values.</p>
<p>(a) x = 2 and y - 3 = (1/2)(x - 2)</p>
<p>(b) x = 2 and 4y + 3x = 18</p>
<p>(c) x = 2 and y - 3 = (-3/4)(x - 2)</p>
<p>(d) Other lines</p>
Step-by-Step Solution
Key Concept: Apply angle between lines formula using tangent addition formulas to find slopes, then write line equations through given point.
<p><strong>Step 1:</strong> Given slope conditions: \(\tan\alpha = \frac{1}{2}\) and \(\tan\theta = 2\)</p><p><strong>Step 2:</strong> Using slope formula \(m = \tan(\alpha ± \theta)\)</p><p><strong>Step 3:</strong> Calculate: \(\tan(\alpha + \theta) = \frac{\frac{1}{2} + 2}{1 - \frac{1}{2} \cdot 2} = \frac{\frac{5}{2}}{0} = \infty\)</p><p><strong>Step 4:</strong> This gives \(m = \infty\) (vertical line) and \(\tan(\alpha - \theta) = -\frac{3}{4}\)</p><p><strong>Step 5:</strong> Equations of lines: \(x = 2\) and \(y - 3 = -\frac{3}{4}(x - 2)\), which simplifies to \(4y + 3x = 18\)</p><p>∴ Answer is (b, d).</p>
Correct Answer: b