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 $\frac{1}{4}$. 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 ?

• A. A only
• B. C only
• C. All the three
• D. B only

Answer 1

Answer: (D)

Let the line be $y=m x+c$
x-intercept : $-\frac{c}{m}$
y-intercept: $c$
A.M of reciprocals of the intercepts
$\frac{-\frac{ m }{ c }+\frac{1}{ c }}{2}=\frac{1}{4} $
$\Rightarrow 2(1- m )= c$
line $: y=m x+2(1-m)=c$
$\Rightarrow (y-2)-m(x-2)=0$
$\Rightarrow $ line always passes through(2,2)