Question

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two non -null vectors such that $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-2\overrightarrow{b}|$. Then the value of $\frac{|\overrightarrow{a}|}{|\overrightarrow{b}|}$ may be

• A. $\frac{1}{4}$
• B. $\frac{1}{8}$
• C. 1
• D. 2

Answer 1

Answer: (C), (D)

The correct options are
C 1
D 2

$|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-2\overrightarrow{b}|$
Squaring on both sides,
$a^2+b^2+2ab\cos \theta =a^2+4b^2-4ab\cos \theta$
$\cos \theta =\frac{b}{2a}\le 1$
$\therefore b\le 2a$
$\frac{a}{b}\ge \frac{1}{2}$
Hence, the possible answers are 1 and 2 according to options.