Mathematics in Physics

JEE Main level, numerical answer, medium

6 of 121

Question 6 of 121

If a⃗\vec{a} and b⃗\vec{b} makes an angle cos⁡−1(59)\cos ^{-1}\left(\frac{5}{9}\right) with each other, then ∣a⃗+b⃗∣=2∣a⃗−b⃗∣|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}| for ∣a⃗∣=n∣b⃗∣|\vec{a}|=n|\vec{b}| The integer value of n\mathrm{n} is ____

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Answer: 3

cos⁡θ=59a⃗⋅b⃗ab=59a2+b2+2a⃗⋅b⃗=2a2+2b2−4a⃗⋅b⃗6a⃗⋅b⃗=a2+b2\begin{aligned} & \cos \theta=\frac{5}{9} \\ & \frac{\vec{a} \cdot \vec{b}}{a b}=\frac{5}{9} \\ & a^2+b^2+2 \vec{a} \cdot \vec{b}=2 a^2+2 b^2-4 \vec{a} \cdot \vec{b} \\ & 6 \vec{a} \cdot \vec{b}=a^2+b^2\end{aligned}
6×59ab=a2+b2103ab=a2+b2&a=nb103nb2=n2b2+b23n2−10n+3=0n=13 and n=3\begin{aligned} & 6 \times \frac{5}{9} a b=a^2+b^2 \\ & \frac{10}{3} a b=a^2+b^2 \quad \& \quad a=n b \\ & \frac{10}{3} n b^2=n^2 b^2+b^2 \\ & 3 n^2-10 n+3=0 \\ & n=\frac{1}{3} \text { and } n=3\end{aligned}
integer value n=3n=3

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