<p>A man starts from the point \(P(-3, 4)\) and reaches point \(Q(0, 1)\) touching \(x\)-axis at \(R(\alpha, 0)\) such that \(PR + RQ\) is minimum, then \(5|\alpha|\) =</p>
Step-by-Step Solution
Key Concept: Use the reflection principle: to minimize PR + RQ where R lies on the x-axis, reflect P across the x-axis to P'(-3, -4), then the minimum path is the straight line from P' to Q, and R is where this line intersects the x-axis.
<p><strong>Step 1:</strong> Reflect point P(-3, 4) across the x-axis to get P'(-3, -4). By the reflection principle, the minimum value of PR + RQ equals P'Q, achieved when P', R, and Q are collinear.</p><p><strong>Step 2:</strong> Find the equation of line through P'(-3, -4) and Q(0, 1).</p><p>Slope = (1 - (-4))/(0 - (-3)) = 5/3</p><p>Equation: y - 1 = (5/3)(x - 0) → y = (5/3)x + 1</p><p><strong>Step 3:</strong> Find point R where this line intersects the x-axis (where y = 0).</p><p>0 = (5/3)x + 1</p><p>(5/3)x = -1</p><p>x = -3/5</p><p>Therefore, α = -3/5</p><p><strong>Step 4:</strong> Calculate 5|α|.</p><p>5|α| = 5|-3/5| = 5 · 3/5 = 3</p><p>∴ Answer: <strong>3</strong></p>
Correct Answer: 3