Straight Lines

The intercept form of a line is x/a + y/b = 1, where a and b are the x- and y-intercepts; this representation lets us encode the intercept-sum condition directly. We require a + b = 10 and that the point (2, 3) lies on the line, giving 2/a + 3/b = 1. Substituting b = 10 - a yields 2(10 - a) + 3a = a(10 - a), so 20 - 2a + 3a = 10a - a^2, which simplifies to a^2 - 9a + 20 = 0, factoring as (a - 4)(a - 5) = 0. Taking a = 4 gives b = 6, producing x/4 + y/6 = 1, i.e. 3x + 2y = 12. The choice 3x + 2y = 10 fails the point test since 6 + 6 ≠ 10. The line 2x + 3y = 13 has intercepts 13/2 and 13/3 summing to about 10.8, not 10. The line x + y = 6 passes through (2, 3) but has intercept sum 12, violating the requirement. This matches the standard JEE Advanced intercept-form technique. Plausibility check: substituting (2, 3) into 3x + 2y gives 6 + 6 = 12, and intercepts 4 and 6 add to exactly 10, confirming both conditions.