On which of the following lines lies the point

Question:

On which of the following lines lies the point

of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$

and the plane, $x+y+z=2$ ?

 

  1. $\frac{x-2}{2}=\frac{y-3}{2}=\frac{z+3}{3}$

  2. $\frac{x-4}{1}=\frac{y-5}{1}=\frac{z-5}{-1}$

  3. $\frac{x-1}{1}=\frac{y-3}{2}=\frac{z+4}{-5}$

  4. $\frac{x+3}{3}=\frac{4-y}{3}=\frac{z+1}{-2}$

Correct Option: , 3

Solution:

General point on the given line is

$x=2 \lambda+4$

$y=2 \lambda+5$

$z=\lambda+3$

Solving with plane,

$2 \lambda+4+2 \lambda+5+\lambda+3=2$

$5 \lambda+12=2$

$5 \lambda=-10$

$\lambda=-2$

$\therefore$ Option $(3)$

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