Question

The straight line
$r=(\hat{i}+\hat{j}+\hat{k})+\alpha (2\hat{i}-\hat{j}+4\hat{k})$ meets the xy-plane at the point

• A. $(2,-1,0)$
• B. $(3,4,0)$
• C. $(\frac{1}{2},\frac{3}{4},0)$
• D. $(\frac{1}{2},\frac{7}{4},0)$
• E. $(\frac{1}{2},\frac{5}{4},0)$

Answer 1

The correct option is E $(\frac{1}{2},\frac{5}{4},0)$

Given, equation of straight line
$r=(\hat{i}+\hat{j}+\hat{k})+\alpha (2\hat{i}-\hat{j}+4\hat{k})$
$\frac{x-1}{2}=\frac{y-1}{-1}=\frac{z-1}{4}=\lambda$
$\Rightarrow (x,y,z)=(2\lambda +1,-\lambda +1,4\lambda +1)$
$\because$ Line meets the $XY$ plane
$\therefore z=0\Rightarrow 4\lambda +1=0$
$\Rightarrow \lambda =-\frac{1}{4}$
$\therefore (x,y,z)=[2\times (-\frac{1}{4})+1,\left(\frac{1}{4}\right)+1,4\times (-\frac{1}{4})\times +1]=(\frac{1}{2},\frac{5}{4},0)$