Equation of a line in intercept form

If a line cuts the coordinate axes at intercepts $a$ and $b$, then its equation is $\frac{x}{a} + \frac{y}{b} = 1.$

Use the given condition

The reciprocals of the intercepts are $\frac{1}{a}$ and $\frac{1}{b}$.

Their arithmetic mean is given as $\frac14$: $\frac{1}{2}\left(\frac{1}{a} + \frac{1}{b}\right) = \frac14.$

So, $\frac{1}{a} + \frac{1}{b} = \frac12.$

Rewrite the line equation

From $\frac{x}{a} + \frac{y}{b} = 1,$ we can write $x\left(\frac{1}{a}\right) + y\left(\frac{1}{b}\right) = 1.$

Since the given points are all of the form $(t,t)$, let us check when $x=y=t$ lies on the line: $t\left(\frac{1}{a} + \frac{1}{b}\right) = 1.$

Using $\frac{1}{a} + \frac{1}{b} = \frac12,$ we get $t\cdot \frac12 = 1 \,\Rightarrow\, t=2.$

Check the stones

• $A=(1,1)$: here $t=1$, not equal to $2$ $\Rightarrow$ not on the line.
• $B=(2,2)$: here $t=2$ $\Rightarrow$ on the line.
• $C=(4,4)$: here $t=4$, not equal to $2$ $\Rightarrow$ not on the line.

Conclusion

Only the stone at $(2,2)$ lies on the man's path.

Therefore, the correct option is D: B only.