Straight Lines
Angle between lines
Grade 11
Question:
<p>Given that the equation \(\sqrt{3}x + y = 1\) has slope angle 120°, any line with inclination of 60° with the above line has slope angle 60°. The equation of such a line passing through the point (3, −2) is:</p>
<p>\(y - \sqrt{3}x + 3\sqrt{3} + 2 = 0\)</p>
<p>\(y - \sqrt{3}x - 3\sqrt{3} + 2 = 0\)</p>
<p>\(y - \sqrt{3}x + 3\sqrt{3} - 2 = 0\)</p>
<p>\(y + \sqrt{3}x - 3\sqrt{3} + 2 = 0\)</p>
Step-by-Step Solution
Key Concept: The slope angle (inclination) of a line is the angle it makes with the positive x-axis. When two lines make an angle of 60° with each other, their slope angles differ by 60°. If one line has slope angle 120°, the other has slope angle either 60° or 180°.
<p><strong>Step 1:</strong> Find the slope angle of the given line √3x + y = 1.</p><p>Rewrite as: y = -√3x + 1, so slope m₁ = -√3 = tan(120°)</p><p>Therefore, the slope angle is 120°.</p><p><strong>Step 2:</strong> Find slope angles of lines inclined at 60° to this line.</p><p>If the angle between two lines with slope angles θ₁ and θ₂ is 60°, then:</p><p>|θ₁ - θ₂| = 60° or θ₁ - θ₂ = ±60°</p><p>With θ₁ = 120°: θ₂ = 120° - 60° = 60° or θ₂ = 120° + 60° = 180°</p><p><strong>Step 3:</strong> The problem states the slope angle is 60°.</p><p>So m₂ = tan(60°) = √3</p><p><strong>Step 4:</strong> Write equation of line with slope √3 passing through (3, -2).</p><p>Using point-slope form: y - (-2) = √3(x - 3)</p><p>y + 2 = √3x - 3√3</p><p>y = √3x - 3√3 - 2</p><p>Or: √3x - y - (3√3 + 2) = 0</p><p>∴ Answer: A</p>
Correct Answer: A