Question

The line joining the points $(2,1,3)$ and $(4,-2,5)$ cuts the plane $2x+y-z=3$.

What is the ratio in which the plane divides the line?

• A. $1:1$
• B. $2:3$
• C. $3:4$
• D. None of the above

Answer 1

The correct option is D None of the above

Equation of line joining two points A=(2,1,3) and B=(4,-2,5),

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

$\frac{x-2}{2}=\frac{y-1}{-3}=\frac{z-3}{2}=\lambda .............\left(1\right)$

let general point P lies on plane $2x+y-z=3$

$P=(2\lambda +2,-3\lambda +1,2\lambda +3)$

Since point P lies on plane $2x+y-z=3$

$2(2\lambda +2)+(-3\lambda +1)-(2\lambda +3)=3$

$4\lambda +4-3\lambda +1-2\lambda -3=3$

$\lambda =-1$

Therefore point P is,

$P=(0,4,1)$

Assume point P cuts AB in the ratio m:n,

By Section formula,

$(\frac{4m+2n}{m+n},\frac{-2m+n}{m+n}\frac{5m+3n}{m+n})=(0,4,1)$

Taking X-coordinate,

$\frac{4m+2n}{m+n}=0$

$4m+2n=0$

$\frac{m}{n}=\frac{-1}{2}$

Therefore P divides AB in the ratio 1:2 (externally)

option (D) is correct.