Point A divides the line segment BC internally in the ratio 4 : 1. If the coordinates of B are (6, 1) and the coordinates of C are (7/2, 6), then what are the exact coordinates of point A? Use the internal section formula and keep the final coordinates in simplest form.

Introduction / Context: This coordinate-geometry problem tests the section formula (internal division). When a point divides a line segment joining two points in a given ratio internally, its coordinates are a weighted average of the endpoints. The key is to interpret the ratio correctly and apply the formula separately to the x-coordinate and y-coordinate. Because the given point C contains a fraction (7/2), careful arithmetic is required to avoid errors. These questions are common in aptitude exams because they check both conceptual understanding (how ratios work on a segment) and accuracy with fractions.

Given Data / Assumptions:

• B = (6, 1) • C = (7/2, 6) • A divides BC internally in the ratio 4 : 1 (BA : AC = 4 : 1) • Required: coordinates of A

Concept / Approach: Internal section formula: If A divides B(x1, y1) and C(x2, y2) in ratio m:n internally (BA:AC = m:n), then Ax = (m*x2 + n*x1) / (m + n) Ay = (m*y2 + n*y1) / (m + n) Here m = 4 and n = 1, so A is closer to C because the weight on C is larger.

Step-by-Step Solution: 1) Identify m = 4, n = 1, B(6, 1), C(7/2, 6). 2) Compute x-coordinate: Ax = (4*(7/2) + 1*6) / 5 Ax = (14 + 6) / 5 = 20/5 = 4 3) Compute y-coordinate: Ay = (4*6 + 1*1) / 5 Ay = (24 + 1) / 5 = 25/5 = 5 4) Therefore A = (4, 5).

Verification / Alternative check: Since BA:AC = 4:1, A should lie closer to C than to B. C has y = 6 and B has y = 1, so A having y = 5 is closer to 6, which matches the ratio intuition. Also, substituting into the weighted-average formula confirms the exact result without approximation.

Why Other Options Are Wrong: • (4, 3) and (3, 5): come from mixing up x and y calculations or using the wrong weights. • (5, 4): often results from reversing B and C in the formula. • (2, 5): indicates an arithmetic mistake while handling 7/2.

Common Pitfalls: • Reversing the ratio (using 1:4 instead of 4:1). • Forgetting that the larger weight applies to the opposite endpoint in the numerator (m multiplies C when ratio is BA:AC = m:n). • Errors while simplifying 4*(7/2).

Final Answer: (4, 5)