Straight Lines
Parallel lines and intercepts
Grade 11
Question:
<p>The equations of lines passing through point \((2, 3)\) and having an intercept of length 2 units between the lines \(2x + y = 3\) and \(2x + y = 5\) are:</p>
<p>(a) \(y = 3\)</p>
<p>(b) \(x = 2\)</p>
<p>(c) \(y = x + 1\)</p>
<p>(d) \(4y + 3x = 18\)</p>
Step-by-Step Solution
Key Concept: The intercept length between two parallel lines cut by a transversal depends on the angle of the transversal and the distance between the parallel lines.
<p>The distance between parallel lines \(2x + y = 3\) and \(2x + y = 5\) is \(\frac{|5-3|}{\sqrt{4+1}} = \frac{2}{\sqrt{5}}\). A line through \((2, 3)\) with intercept 2 between these lines must be perpendicular to them with proper spacing, or be parallel to them. The candidates are \(y = 3\), \(x = 2\), and \(4y + 3x = 18\).</p>
Correct Answer: a, b, d