There is indeed a formula for \(a\) and \(b.\) However, this is general problem and there is no opportunity for any helpful tricks like there were in the Module 4 Day 13 Challenge problem.

To find the formula for \(a\) and \(b\) in this problem, you will have to use a geometric coordinate system. Here is an algorithm for how to do it:

**Note:** Let the point \(M\) be a midpoint of a segment \(\overline{AB}.\) The \(\triangle AB\text{(meeting point)}\) is a right isosceles triangle and \(M\) is a midpoint, so the distance from the point \(M\) to a meeting point (which is a median and height length of right isosceles \(\triangle\)) is equal to half of \(AB\) (which is the hypotenuse of the right isosceles \(\triangle\)).

**Step 1:** Find an equation of the line \(AB\) (where \(A\) and \(B\) are starting positions of Alice and Bob);

**Step 2:** Find the coordinates of point \(M;\)

**Step 3:** Find an equation of the line perpendicular to \(AB\) in point \(M\) using the equation of line \(AB\) and coordinates of point \(M;\)

**Step 4:** Find the length of segment \(\overline{AB};\)

**Step 5:** Measure out the length of a half of segment \(\overline{AB}\) on perpendicular to \(AB\) in point \(M\) and find the coordinates of the obtained point. This coordinate will give us our \(a\) and \(b.\)

If you are interesting in finding the formula for \(a\) and \(b,\) you can try to do it yourself using this algorithm. However, it would be pretty complicated if you were not familiar to geometric coordinate system before.

So, once again, there exists a formula for \(a\) and \(b,\) but in this general problem there is no way to employ any helpful tricks in order to make it easier.

It is great that you a trying to analyze and think over the problems that you see!

The Module 4 Day 13 Challenge problem was very well created by Prof. Po and the numbers there are pretty nice ðŸ™‚

To begin with, if you are changing the angle between the directions where Alice and Bob are came from to the angle that are different from \(90^{\circ}\) (as well as \(180^{\circ}\)) it will be hard to find any tricks or nice answers if the other numbers in problem are not extremely nice and beautiful. So to solve this problem with random numbers will be complicated task.

To solve it you will need to use geometric coordinate system and trigonometry together. You will learn how to do it in more advance classes.