Can you show us what you have tried?
Hint: If X and Y are independent then P(X = x AND Y = y) = P(X = x)*P(Y = y). where P(X = x) = e^(-x) and similarly for P(Y = y).
Now you need to consider the region for Y < tX. This means you will need to look at the region of integration where this happens. Since t > 0 is given and since this is a linear relationship, if you use a double integral then you can find the region under the triangle.
If X = 0 then Y = 0. If X is infinity then Y is infinity as well. Now you will get a line from 0 to infinity with a gradient equal to t and the region will be between the x-axis and this line.
Using this hint, can you construct an integral with the right integration region to get your answer?