Q 11-03-074JEE MainJEE Main 2024 (9 Apr, Shift 1)Medium
If $\vec a$ and $\vec b$ make an angle $\cos^{-1}\left(\dfrac59\right)$ with each other, then $|\vec a + \vec b| = \sqrt2\,|\vec a - \vec b|$ for $|\vec a| = n|\vec b|$. The integer value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 3
Squaring, with $\cos\theta = \dfrac59$:
$$a^2 + b^2 + 2ab\cos\theta = 2\left(a^2 + b^2 - 2ab\cos\theta\right) \;\Rightarrow\; a^2 + b^2 = 6ab\cos\theta = \frac{10}{3}ab$$
With $a = nb$: $3n^2 - 10n + 3 = 0 \Rightarrow n = 3$ or $\dfrac13$. The integer value is $n = 3$.
Solution by Sreeraj P, M.Sc Physics