The correct answer is $\boxed{{-\frac{{{P_L}}}{{{P_K}}}}}$.
An isocost curve is a line that shows all the combinations of capital and labor that can be purchased with a given amount of money. The slope of the isocost curve is equal to the negative of the ratio of the prices of capital and labor. This is because the amount of capital that can be purchased with a given amount of money is inversely proportional to the price of capital, and the amount of labor that can be purchased with a given amount of money is inversely proportional to the price of labor.
For example, if the price of capital is $10 per unit and the price of labor is $5 per unit, then a firm with $100 to spend on inputs can purchase 10 units of capital or 20 units of labor. The isocost curve for this firm would be a straight line with a slope of $-\frac{1}{2}$.
The isocost curve is a useful tool for analyzing the cost of production. It can be used to determine the least-cost combination of inputs to produce a given level of output, or to determine the maximum level of output that can be produced with a given budget.