<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: The maximum value of |MA - MB| occurs when M lies on the line segment extended such that M is collinear with A and B, specifically on the line AB extended beyond one of the points. Since M must lie on 2x + 3y = 5, find the intersection of line AB with this given line.
<p><strong>Step 1:</strong> Find the equation of line AB passing through A(2, 3) and B(1, 2).</p><p>Slope = (3 - 2)/(2 - 1) = 1</p><p>Line AB: y - 2 = 1(x - 1) ⟹ y = x + 1 or x - y + 1 = 0</p><p><strong>Step 2:</strong> For |MA - MB| to be maximum, points M, A, B must be collinear (by triangle inequality: |MA - MB| ≤ |AB| with equality when collinear).</p><p><strong>Step 3:</strong> Find intersection of line AB (x - y + 1 = 0) with the given line (2x + 3y = 5).</p><p>From x - y + 1 = 0: x = y - 1</p><p>Substitute into 2x + 3y = 5:</p><p>2(y - 1) + 3y = 5</p><p>2y - 2 + 3y = 5</p><p>5y = 7</p><p>y = 7/5, so x = 7/5 - 1 = 2/5</p><p><strong>Step 4:</strong> Thus M(h, k) = (2/5, 7/5).</p><p>Calculate 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