Question

If the line $\frac{2x-1}{2}=\frac{2-y}{2}=\frac{z+1}{k}$ is parallel to the plane $2x-y+z=3,$ then k = _______________.

Answer 1

$\mathrm{Given}\mathrm{line}\frac{2x-1}{2}=\frac{2-y}{2}=\frac{z+1}{k}\mathrm{is}\mathrm{parallel}\mathrm{to}\mathrm{plane}2x-y+z=3$
Line is of the pr $\frac{x-{\displaystyle \frac{1}{2}}}{1}=\frac{y-2}{-2}=\frac{z-\left(-1\right)}{k}$

is parallel if $\overrightarrow{n}$for plane is perpendicular to line for plane 2x – y + z = 3

$$\begin{gathered} \mathrm{Normal}\mathrm{is}\overrightarrow{n}2\hat{i}-\hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+k\hat{k} \\ \mathrm{Since}\overrightarrow{n}.\overrightarrow{b}=0 \\ \mathrm{i}.\mathrm{e}.2\left(1\right)-1\left(-2\right)+1\left(k\right)=0 \\ \mathrm{i}.\mathrm{e}.2+2+k=0 \\ \mathrm{i}.\mathrm{e}.k=-4 \end{gathered}$$