Straight Lines
Intersection of lines
Grade 11
Question:
<p>The line parallel to the <i>x</i>-axis and passing through the intersection of the lines \(2by + 3b = 0\) and \(hx - 2ay - 3a = 0\), where \((a, b) \neq (0, 0)\) is</p>
<p>below the <i>x</i>-axis at a distance of \(\dfrac{3}{2}\) from it.</p>
<p>below the <i>x</i>-axis at a distance of \(\dfrac{2}{3}\) from it.</p>
<p>below the <i>x</i>-axis at a distance of \(\dfrac{3}{2}\) from it.</p>
<p>above the <i>x</i>-axis at a distance of \(\dfrac{2}{3}\) from it.</p>
Step-by-Step Solution
Key Concept: A line parallel to the x-axis has the form y = c (constant). Find the intersection point of the two given lines, then extract the y-coordinate, which gives the equation of the required line.
<p><strong>Step 1:</strong> Analyze the first equation: 2by + 3b = 0</p><p>Factor out b (given b ≠ 0): b(2y + 3) = 0</p><p>Therefore: 2y + 3 = 0, which gives y = -3/2</p><p><strong>Step 2:</strong> Verify this satisfies the second equation at the intersection.</p><p>The second equation hx - 2ay - 3a = 0 must also pass through the intersection point. Since any point on the intersection satisfies both equations, the y-coordinate we found is the y-coordinate of intersection.</p><p><strong>Step 3:</strong> Write the line parallel to the x-axis.</p><p>A line parallel to the x-axis passing through y = -3/2 is:</p><p><strong>y = -3/2</strong> or equivalently <strong>2y + 3 = 0</strong></p><p>∴ Answer: A</p>
Correct Answer: A