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Q.
The equation of the projection line of the line $\frac{x + 1}{2}=\frac{y + 1}{- 1}=\frac{z + 3}{4}$ on the plane $x+2y+z=6$ is
NTA AbhyasNTA Abhyas 2022
Solution:
From the diagram,
Coordinates of the point $A$ are $\left(2 \lambda - 1 , - \lambda - 1 , 4 \lambda - 3\right)$ (from the line)
It must satisfy the equation of the plane
$2\lambda -1-2\lambda -2+4\lambda -3=6\Rightarrow \lambda =3$
So, coordinates of the point $A$ are $\left(5 , - 4,9\right)$
Point $B^{'}$ must be the foot of perpendicular for point $B$
$\frac{x + 1}{1}=\frac{y + 1}{2}=\frac{z + 3}{1}=-\frac{\left(- 1 - 2 - 3 - 6\right)}{1 + 4 + 1}=2$
$x=1,y=3,z=-1$
Direction ratios of $ < AB^{'}>= < 4,-7,10>$
So, the equation of projection of line is $\frac{x - 1}{4}=\frac{y - 3}{- 7}=\frac{z + 1}{10}$
