A man is walking on a straight line.

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 $${1 \over 4}$$. Three stones A, B and C are placed at the points
(1, 1), (2, 2) and (4, 4) respectively. Then, which of these stones is / are on the path of the man?

Accepted Answer

Given, position of A = (1, 1)

Position of B = (2, 2)

Position of C = (4, 4)

Let x-intercept be a and y-intercept be b.

Equation of line traced is

$${x \over a} + {y \over b} = 1$$

This is the equation of path, let a point (h, k) lie on this path.

Then, $${h \over a} + {k \over b} = 1$$

Also, AM of reciprocal of a and b = $${1 \over 4}$$

$$\therefore$$ $${{{1 \over a} + {1 \over b}} \over 2} = {1 \over 4}$$

$${1 \over a} + {1 \over b} = {1 \over 2}$$

On comparing Eqs. (i) and (ii), we get (h, k) = (2, 2)

Hence, the required stone is B(2, 2).

Solutions · 1