First slide

Means - A.M/G.M/H.M

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?

Moderate

A

B only
B

A only
C

C only
D

All the three

Solution

$\begin{array}{l} \frac{x}{a} + \frac{y}{b} = 1 \\ \frac{h}{a} + \frac{k}{b} = 1 - - - - (1) \\ \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{1}{4} \\ ∴ \frac{1}{a} + \frac{1}{b} = \frac{1}{2} - - - - - (2) \end{array}$
$∴ Line passes through fixed point (2, 2)$ . $(from (1) and (2))$

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