Q71 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

If $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ are vectors such that $|\vec{a}+\vec{b}|=25$, then the sum of the values of $t$ is

  • A
    $14$
  • B
    $11$
  • Correct answer
    C
    $4$
  • D
    $77$

Solution

Given, $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}, \hat{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ $\begin{aligned} & \text { Now, }|\vec{a}+\vec{b}|=25 \\ & \Rightarrow|(t-2) \hat{i}+7 \hat{j}+0 \hat{k}|=25 \Rightarrow \sqrt{(t-2)^2+49+0^2}=25 \\ & \Rightarrow(t-2)^2+49=625 \Rightarrow(t-2)^2=576 \\ & \Rightarrow t^2-4 t+4=576 \end{aligned}$ So sum of value of $t=4$

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

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