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  • If the lines <img alt="\mathrm{\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3 \hat{j}-\hat{k})}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cvec%7Br%7D%3D%28%5Chat%7Bi%7D-%5Cha

If the lines \mathrm{\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3 \hat{j}-\hat{k})} and \mathrm{\vec{r}=(\alpha \hat{i}-\hat{j})+\mu(2 \hat{i}-3 \hat{k})} are coplanar, then the distance of the plane containing these two lines from the point \mathrm{(\alpha, 0,0)} is:

Option: 1

\frac{2}{9}


Option: 2

\frac{2}{11}


Option: 3

\frac{4}{11}


Option: 4

2


Answers (1)

best_answer

lines ort co plonar.
\left|\begin{array}{ccc} 1-2 & 0 & 1 \\ 0 & 3 & -1 \\ 2 & 0 & -3 \end{array}\right|=0

\begin{aligned} &(1-\alpha)(-5)-6=0 \\ &\alpha-1=\frac{2}{3} \quad \Rightarrow \quad \alpha=\frac{5}{3} \end{aligned}
Nermal vector of plans: (-9,-2,-6)
\therefore Equation of plans: \mathrm{-9(x-1)-2(y+1)-6(2-1)=0}
\mathrm{9 x+2 y+6 z-13=0 }
\therefore Distance from \left(\frac{5}{3}, 0,0\right)=\frac{15-13}{\sqrt{{81+4+36}}}=\frac{2}{11}

Posted by

Divya Prakash Singh

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