$\mathrm{Foot}\mathrm{of}\mathrm{perpendicular}\mathrm{from}\mathrm{origin}\mathrm{on}a\mathrm{line}\mathrm{passing}\mathrm{through}(1,1,1)\mathrm{having}\mathrm{direction}\mathrm{ratios}<2,3,4>,is$

The correct option is:
$(\frac{11}{29},\frac{2}{29},-\frac{7}{29})$

Step-by-Step Derivation:

Step 1: Write the equation of the line.
The equation of a line passing through a point $(x_1,y_1,z_1)$ with direction ratios $<a,b,c>$ is given by:
$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Substituting the given point $(1,1,1)$ and direction ratios $<2,3,4>$, we get:
$\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{4}=\lambda$

Step 2: Express a general point on the line.
Any general point $P$ on this line can be written in terms of parameter $\lambda$ as:
$P=(2\lambda +1,3\lambda +1,4\lambda +1)$

Step 3: Relate the line to the origin.
Let $P$ be the foot of the perpendicular from the origin $O(0,0,0)$ to the line.
The direction ratios of the line segment $OP$ are:
$(2\lambda +1-0,3\lambda +1-0,4\lambda +1-0)=(2\lambda +1,3\lambda +1,4\lambda +1)$

Step 4: Use the perpendicularity condition.
Since $OP$ is perpendicular to the given line (which has direction ratios $<2,3,4>$), their dot product must be zero:
$2(2\lambda +1)+3(3\lambda +1)+4(4\lambda +1)=0$

Step 5: Solve for $\lambda$.
Expanding and simplifying the equation:
$4\lambda +2+9\lambda +3+16\lambda +4=0$
$29\lambda +9=0$
$\lambda =-\frac{9}{29}$

Step 6: Calculate coordinates of the foot of the perpendicular.
Substitute the value of $\lambda$ back into the coordinates of point $P$:
For the x-coordinate:
$x=2(-\frac{9}{29})+1=-\frac{18}{29}+\frac{29}{29}=\frac{11}{29}$
For the y-coordinate: