Consider the following statements : There are six villages A, B, C, D, E and F. F is 1 km to the west of D. B is 1 km to the east of E. A is 2 km to the north of E. C is 1 km to the east of A. D is 1 km to the south of A. Which three villages are in a line ?

(a) A, C, B
(b) A, D, E ✓ UPSC's answer
(c) C, B, F
(d) E, B, D

Why the answer is (b)

• Take E as (0, 0), so B is (1, 0) because B is 1 km east of E.

• A is (0, 2) because A is 2 km north of E, and D is (0, 1) because D is 1 km south of A.

• C is (1, 2) because C is 1 km east of A, and F is (-1, 1) because F is 1 km west of D.

• A (0, 2), D (0, 1), and E (0, 0) all have x-coordinate 0, so they lie on the same north-south line.

• Therefore option (b) A, D, E is the correct line.

Why the other options are wrong

(a) A, C, B: A (0, 2), C (1, 2), and B (1, 0) do not share a common straight line because A-C is horizontal and C-B is vertical.
(c) C, B, F: C (1, 2), B (1, 0), and F (-1, 1) are not collinear because F is not on the vertical line x = 1 through C and B.
(d) E, B, D: E (0, 0), B (1, 0), and D (0, 1) are not collinear because D is not on the horizontal line y = 0 through E and B.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2014, held on 24 August 2014. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.