<p>The distance of the point (1, 2) from the line \(x + y + 5 = 0\) measured along the line parallel to \(3x - y = 7\) is equal to:</p>
Step-by-Step Solution
Key Concept: To find distance along a direction, first find the point where the perpendicular from (1,2) to a line parallel to 3x - y = 7 meets that line, then calculate the distance between these two points.
<p><strong>Step 1:</strong> Find the line through (1, 2) parallel to 3x - y = 7.</p><p>The slope of 3x - y = 7 is 3. So the line through (1, 2) with slope 3 is:<br/>y - 2 = 3(x - 1)<br/>⟹ y = 3x - 1 or 3x - y - 1 = 0</p><p><strong>Step 2:</strong> Find the intersection of 3x - y - 1 = 0 with x + y + 5 = 0.</p><p>From 3x - y - 1 = 0: y = 3x - 1<br/>Substitute in x + y + 5 = 0:<br/>x + (3x - 1) + 5 = 0<br/>4x + 4 = 0<br/>x = -1, y = -4</p><p><strong>Step 3:</strong> Find the distance between (1, 2) and (-1, -4).</p><p>Distance = √[(1-(-1))² + (2-(-4))²]<br/>= √[4 + 36]<br/>= √40<br/>= 2√10</p><p>∴ Answer: A (2√10)</p>
Correct Answer: A