Question

The foot of perpendicular from $A(1,1,1)$ to the line joining $B(-8,5,6)$ and $C(12,1,0)$ lies on the plane(s)

• A. $2x+y+z=10$
• B. $x-y+z=2$
• C. $x+y-z=2$
• D. $x+y+z=8$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $2x+y+z=10$
B $x-y+z=2$
C $x+y-z=2$
D $x+y+z=8$


Equation of BC
$\overrightarrow{r}=(-8\hat{i}+5\hat{j}+6\hat{k})+\lambda (20\hat{i}-4\hat{j}-6\hat{k})\overrightarrow{r}=(-8+20\lambda )\hat{i}+(5-4\lambda )\hat{j}+(6-6\lambda )\hat{k}$
Coordinates of $M$ is $(-8+20\lambda ,5-4\lambda ,6-6\lambda )$
$\overrightarrow{AM}.\overrightarrow{BC}=020(-9+20\lambda )-4(4-4\lambda )-6(5-6\lambda )=0\lambda =\frac{1}{2}$
So $M\to (2,3,3)$