Straight Lines

The area of a triangle from its vertices is computed by the determinant or shoelace formula, which gives a signed area whose magnitude is the geometric area. Using area = (1/2)|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|, substitute A(1, 2), B(4, 6), C(-3, 5). This gives (1/2)|1(6 - 5) + 4(5 - 2) + (-3)(2 - 6)| = (1/2)|1(1) + 4(3) + (-3)(-4)| = (1/2)|1 + 12 + 12| with the absolute bracket evaluating to 25, so the area is 25/2 square units. Option 29/2 adds a stray contribution to the bracket. Option 15 over-rounds the half. Option 12 drops a term entirely. This is the standard JEE Advanced coordinate-area technique. Plausibility check: all three points are well separated, so an area near 12 to 13 square units is reasonable, and the determinant being nonzero confirms the points are non-collinear.