Solve The Following Questions


The foot of the perpendicular drawn from the point $(4,2,3)$ to the line joining the points

$(1,-2,3)$ and $(1,1,0)$ lies on the plane :

  1. $x+2 y-z=1$

  2. $x-2 y+z=1$

  3. $x-y-2 z=1$

  4. $2 x+y-z=1$

Correct Option: , 4


Equation of $\mathrm{AB}=\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}+\hat{\mathrm{j}})+\lambda(3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$

Let coordinates of M $=(1,(1+3 \lambda),-3 \lambda)$.

$\overrightarrow{\mathrm{PM}}=-3 \hat{\mathrm{i}}+(3 \lambda-1) \hat{\mathrm{j}}-3(\lambda+1) \hat{\mathrm{k}}$

$\overline{\mathrm{AB}}=3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$

$\Rightarrow \lambda=-\frac{1}{3}$

$\therefore \quad M=(1,0,1)$

Clearly $\mathrm{M}$ lies on $2 \mathrm{x}+\mathrm{y}-\mathrm{z}=1$.

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