Q71 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

If $|\bar{a}|=4,|\bar{b}|=5,|\bar{a}-\bar{b}|=3$ and $\theta$ is the angle between the vectors $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$, then $\cot ^2 \theta=$

  • Correct answer
    A
    $\frac{9}{16}$
  • B
    $\frac{4}{3}$
  • C
    $\frac{3}{4}$
  • D
    $\frac{16}{9}$

Solution

$|\vec{a}|=4,|\vec{b}|=5$ $\begin{aligned} & |\vec{a}-\vec{b}|=3 \Rightarrow|\vec{a}-\vec{b}|^2=9 \\ & (\vec{a}-\vec{b}) \cdot(\vec{a}-\vec{b})=9 \\ & |\vec{a}|^2-2|\vec{a} \| \vec{b}| \cos \theta+|\vec{b}|^2=9 \\ & 16+25-2(4)(5) \cos \theta=9 \\ & \cos \theta=\frac{4}{5} \Rightarrow \cot \theta=\frac{4}{3} ; \cot ^2 \theta=\frac{16}{9}\end{aligned}$

Hence, the correct answer is option: 'A'.

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