$\text{ A line through the variable point}\mathrm{A}(\mathrm{k}+1,2\mathrm{k})\text{meets the lines}7\mathrm{x}+\mathrm{y}-16=0,5\mathrm{x}-\mathrm{y}-8=0\text{and}x-5y+8=0$$\text{ at }B,C,D\text{ respectively. Then }AC,AB,AD\text{ are in }\dots$

$$\begin{gathered} \text{ Let the equation of line passing through }\mathrm{A}(\mathrm{k}+1,2\mathrm{k})\text{ making an angle }\theta \text{ with the }+\mathrm{ve} \\ \text{ direction of }\mathrm{x}-\text{ axis be }\frac{\mathrm{x}-(\mathrm{k}+1)}{\cos \mathrm{\theta }}=\frac{\mathrm{y}-2\mathrm{k}}{\sin \mathrm{\theta }}={\mathrm{r}}_1,{\mathrm{r}}_2,{\mathrm{r}}_3 \\ \text{ (if }\mathrm{AB}={\mathrm{r}}_1,{\mathrm{r}}_2,{\mathrm{r}}_3\text{ ) (if }\mathrm{AB}={\mathrm{r}}_1,\mathrm{AC}={\mathrm{r}}_2,\mathrm{AD}={\mathrm{r}}_3\text{ ) } \\ \therefore \mathrm{B}\equiv \left[(\mathrm{k}+1)+{\mathrm{r}}_1\cos \mathrm{\theta },2\mathrm{k}+{\mathrm{r}}_1\sin \mathrm{\theta }\right] \\ \mathrm{C}\equiv \left[(\mathrm{k}+1)+{\mathrm{r}}_2\cos \mathrm{\theta },2\mathrm{k}+{\mathrm{r}}_2\right.\sin \mathrm{\theta }] \\ \mathrm{D}\equiv \left[(\mathrm{k}+1)+{\mathrm{r}}_3\cos \mathrm{\theta },2\mathrm{k}+{\mathrm{r}}_3\sin \mathrm{\theta }\right] \\ \text{ Hence }{\mathrm{r}}_2,{\mathrm{r}}_1,{\mathrm{r}}_3\text{ are in H.P. } \end{gathered}$$