Question:

<p><strong>Step 1:</strong> For a line with x-intercept a and y-intercept b, the intercept form is x/a + y/b = 1. The area of the triangle formed with coordinate axes is |ab|/2 = 12, so |ab| = 24.</p><p><strong>Step 2:</strong> Since the line passes through (2,3), substituting into the intercept form: 2/a + 3/b = 1.</p><p><strong>Step 3:</strong> From |ab| = 24, we have ab = ±24. Let's consider both cases.</p><p><strong>Case 1: ab = 24</strong></p><p>From 2/a + 3/b = 1, we get: 2b + 3a = ab = 24</p><p>So: 2b + 3a = 24, which gives b = (24-3a)/2</p><p>Substituting into ab = 24: a·(24-3a)/2 = 24</p><p>24a - 3a² = 48</p><p>3a² - 24a + 48 = 0</p><p>a² - 8a + 16 = 0</p><p>(a-4)² = 0, so a = 4, then b = (24-12)/2 = 6</p><p>One solution: (a,b) = (4,6)</p><p><strong>Case 2: ab = -24</strong></p><p>From 2/a + 3/b = 1, we get: 2b + 3a = ab = -24</p><p>So: 2b + 3a = -24, which gives b = (-24-3a)/2</p><p>Substituting into ab = -24: a·(-24-3a)/2 = -24</p><p>-24a - 3a² = -48</p><p>3a² + 24a - 48 = 0</p><p>a² + 8a - 16 = 0</p><p>a = (-8 ± √(64+64))/2 = (-8 ± √128)/2 = (-8 ± 8√2)/2 = -4 ± 4√2</p><p><strong>Step 4:</strong> This gives us two more values:</p><p>a₁ = -4 + 4√2, then b₁ = -24/a₁ = -24/(-4+4√2) = 6/(√2-1) = 6(√2+1) (after rationalization)</p><p>a₂ = -4 - 4√2, then b₂ = -24/a₂ = -24/(-4-4√2) = 6/(√2+1) = 6(√2-1) (after rationalization)</p><p><strong>Step 5:</strong> We have four valid pairs: (4,6), (-4+4√2, 6(√2+1)), and (-4-4√2, 6(√2-1)), plus verification shows all satisfy both conditions. Each pair represents a distinct line passing through (2,3) with area 12.</p><p><strong>∴ Answer:</strong> D</p>