A book vendor sold a book at a loss of $10\%$. Had he sold it for $\text{₹}1,080$ more, he would have earned a profit of $10\%$. Find out the cost of the book.

Let the Cost Price ($\text{CP}$) of the book be $x$.
Initially, selling at a $10\%$ loss means the selling price is $90\%$ of $x$.
If sold for $\text{₹}1,080$ more, the profit is $10\%$, making the new selling price $110\%$ of $x$.

The difference between the two selling conditions corresponds to the cash increase:

$$\text{Difference in percentage}=110\%-90\%=20\%$$

According to the problem:

$$20\%\text{ of }x=1080$$

$$\frac{20}{100}\times x=1080$$

$$\frac{1}{5}\times x=1080$$

$$x=1080\times 5=5400$$

Therefore, the cost price of the book is $\text{₹}5,400$.