Q70 / 160 · TS EAMCET 2024 (10 May Shift 2) · Mathematics

$\vec{n}$ is a unit vector normal to the plane $\pi$ containing the vectors $\hat{i}+3 \hat{k}$ and $2 \hat{i}+\hat{j}-\hat{k}$. If this plane $\pi$ passes through the point $(-3,7,1)$ and p is the perpendicular distance from the origin to this plane $\pi$, then $\sqrt{p^2+5}=$

  • A
    59
  • Correct answer
    B
    8
  • C
    64
  • D
    51

Solution

Equation of plane $\pi$ containing two vectors $\alpha$ and $\beta$ and passes through point $a$ is $(\vec{r}-\vec{a})(\vec{\alpha} \times \vec{\beta})=0$ $\vec{\alpha} \times \vec{\beta}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 3 \\ 2 & 1 & -1\end{array}\right|=-3 \hat{i}+7 \hat{j}+\hat{k}$ $\begin{aligned} & {[(x+3) \hat{i}+(y-7) \hat{j}+(z-1) \hat{k}][-3 \hat{i}+7 \hat{j}+\hat{k}]=0} \\ & \Rightarrow-3(x+3)+7(y-7)+(z-1)=0 \\ & \Rightarrow-3 x-9+7 y-49+z-1=0 \Rightarrow-3 x+7 y+z-59=0 \\ & p=\left|\frac{0+0+0-59}{\sqrt{9+49+1}}\right|=\sqrt{59} \therefore \sqrt{p^2+5}=8\end{aligned}$

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

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