The number of values of k for which the lines (k + 1)x + 8y = 4k and kx + (k + 3)y = 3k - 1 are coincident is

Lines (k + 1)x + 8y = 4k and kx + (k + 3)y = 3k - 1 are coincident.
Then we can compare the ratio of coefficients as

$\begin{array}{rl} & \frac{\mathrm{k}+1}{\mathrm{k}}=\frac{8}{\mathrm{k}+3}=\frac{4\mathrm{k}}{3\mathrm{k}-1}\\ \mathrm{or} & {\mathrm{k}}^2+4\mathrm{k}+3=8\mathrm{k} \text{and }24\mathrm{k}-8=4{\mathrm{k}}^2+12\mathrm{k}\\ \mathrm{or} & (\mathrm{k}-3)(\mathrm{k}-1)=0 \text{and }(\mathrm{k}-2)(\mathrm{k}-1)=0 \text{or }\mathrm{k}=1\end{array}$