Straight Lines
Distance of a point from a line
Grade 11
Question:
<p>Lines are drawn parallel to the line \(4x - 3y + 2 = 0\), at a distance \(\dfrac{3}{5}\) from the origin. Then which one of the following points lies on any of these lines?</p>
<p>\(\left(-\dfrac{1}{4}, \dfrac{2}{3}\right)\)</p>
<p>\(\left(\dfrac{1}{4}, -\dfrac{1}{3}\right)\)</p>
<p>\(\left(\dfrac{1}{4}, \dfrac{1}{3}\right)\)</p>
<p>\(\left(-\dfrac{1}{4}, -\dfrac{2}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Parallel lines to 4x - 3y + 2 = 0 have the form 4x - 3y + c = 0. Use the distance formula from origin to line: |c|/√(16+9) = 3/5 to find c, then check which point satisfies the resulting equations.
<p><strong>Step 1:</strong> Lines parallel to 4x - 3y + 2 = 0 have the form 4x - 3y + c = 0.</p><p><strong>Step 2:</strong> Distance from origin (0,0) to line 4x - 3y + c = 0 is: d = |c|/√(4² + (-3)²) = |c|/5</p><p><strong>Step 3:</strong> Given that d = 3/5, we have: |c|/5 = 3/5 ⟹ |c| = 3 ⟹ c = 3 or c = -3</p><p><strong>Step 4:</strong> The two parallel lines are: <br>• 4x - 3y + 3 = 0<br>• 4x - 3y - 3 = 0</p><p><strong>Step 5:</strong> Check which given point satisfies either equation by substituting its coordinates into both equations.</p><p><strong>Note:</strong> Without the answer choices provided, verify the correct point by substituting into 4x - 3y ± 3 = 0.</p><p>∴ Answer: A</p>
Correct Answer: A