<p>If the point \(M(h, k)\) lie on the line \(2x + 3y = 5\) such that \(|MA - MB|\) is maximum where \(A(2, 3)\) and \(B(1, 2)\), then find the value of \((3h + 2k)\).</p>
Step-by-Step Solution
Key Concept: To maximize |MA - MB|, point M must lie on the line segment extended through A and B. The maximum occurs when M lies on the line passing through A and B, specifically on the extension beyond the farther point. Use the condition that A, M, B are collinear when |MA - MB| is maximum.
<p><strong>Step 1:</strong> For |MA - MB| to be maximum, by the triangle inequality, points A, M, and B must be collinear. This is because |MA - MB| ≤ |AB| with equality when M lies on line AB extended.</p><p><strong>Step 2:</strong> Find the equation of line through A(2, 3) and B(1, 2):<br/>Slope = (3-2)/(2-1) = 1<br/>Line AB: y - 2 = 1(x - 1) → y = x + 1 or x - y + 1 = 0</p><p><strong>Step 3:</strong> Point M(h, k) lies on both lines:<br/>Line 1: 2h + 3k = 5<br/>Line 2: h - k + 1 = 0 → k = h + 1</p><p><strong>Step 4:</strong> Substitute k = h + 1 into 2h + 3k = 5:<br/>2h + 3(h + 1) = 5<br/>2h + 3h + 3 = 5<br/>5h = 2<br/>h = 2/5</p><p><strong>Step 5:</strong> Find k: k = h + 1 = 2/5 + 1 = 7/5<br/>So M(2/5, 7/5)</p><p><strong>Step 6:</strong> Calculate 3h + 2k:<br/>3h + 2k = 3(2/5) + 2(7/5) = 6/5 + 14/5 = 20/5 = 4</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4