If a and b make an angle cos ⁻¹(5/9) with each other, then |a + b| = √(2)|a - b| for |a| = n|b|. The integer value of n is _________. Numerical answer type. Work it out, then open the answer.

Answer: 3. Square both sides: a² + b² + 2abcos θ = 2(a² + b² - 2abcos θ), so 6abcos θ = a² + b². Put cos θ = 5/9 and a = nb: 10/3n = n² + 1. 3n² - 10n + 3 = 0 ⇒ n = 3 or 1/3. Integer value: n = 3

If a and b make an angle cos ⁻¹(5/9) with each other, then |a + b| =… | JEE Main 2024 (9 Apr, Shift 1)

Q12JEE Main 2024 · 9 APR · SHIFT 1Medium

Addition & Subtraction of Vectors

If $\vec{a}$ and $\vec{b}$ make an angle $\cos^{-1}\left(\frac{5}{9}\right)$ with each other, then $|\vec{a} + \vec{b}| = \sqrt{2}|\vec{a} - \vec{b}|$ for $|\vec{a}| = n|\vec{b}|$ . The integer value of $n$ is _________.

Numerical answer type. Work it out, then open the answer.

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Answer: 3

Square both sides: $a^2 + b^2 + 2ab\cos\theta = 2(a^2 + b^2 - 2ab\cos\theta)$ , so $6ab\cos\theta = a^2 + b^2$ .
Put $\cos\theta = \frac{5}{9}$ and $a = nb$ : $\frac{10}{3}n = n^2 + 1$ .
$3n^2 - 10n + 3 = 0 \Rightarrow n = 3$ or $\frac{1}{3}$ .
Integer value: $n = 3$

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