Question

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is `1/4`. Three stones A, B and C are placed at points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?

Options

• B only

• A only

• All the three

• C only

Answer 1

B only

Explanation:

Given: A man is walking in a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate areas is `1/4`. Three stones A, B and C are placed at points (1, 1), (2, 2) and (4, 4) respectively. The equation of a straight line having intercepts a and b is `x/"a" + "y"/"b"` = 1

This line passes through (h, k) 

∴ `"h"/"a" + "k"/"b"` = 1  ...(i)

A.M. of reciprocals of intercepts `(1/"a" + 1/"b")/2 = 1/4`

⇒ `1/"a" + 1/"b" = 1/2`  ...(ii)

Compare (i) and (ii)

h = 2, k = 2

The line passes through (2, 2).

So, only stone B is on the path of man.