$\begin {array} {1 1} (A)\;k=2 & \quad (B)\;k=-2 \\ (C)\;k=\pm 2 & \quad (D)\;\text{ none of the above } \end {array}$

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- If the given line is parallel to $y$ - axis, then its slope is undefined.

The given equation of the line is

$(k-3)x-(4-k^2)y+k^2-7k+6=0$

It is given that the given line is parallel to $y$ axis.

The above line can be written in the form $y=mx+c$

i.e., $y = \large\frac{(k-3)}{(4-k^2)}$$x+ \large\frac{k^2-7k+6}{(4-k^2)}$

Hence the slope of the given line is

$ \large\frac{k-3}{4-k^2}$

Since it is parallel to $y$ - axis, $ \large\frac{k-3}{4-k^2}$$ = \infty$

(i.e., ) $4-k^2=0$

or $k^2=4$

$k = \pm 2$

Hence the value of $k$ is $ \pm 2$

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