Do we actually take random line in first step of linear regression? 
This is the screenshot I took from a video on linear regression made by Luis Serrano. He explained linear regression step by step (scratch version). The first step was to start with a random line.
The question is do we actually draw a random line, or instead do we perform some calculation like taking an average of y values and initially draw a line. Because if we take any random line it might not fall near any points at all. Maybe it will fall on the 3rd quadrant of the coordinate system where there are no points in this case.

We, sort of, do something like this effectively, especially in Gradient descent algorithms. A random line is simply a set of random parameters $\beta_0,\beta_1$. The gradient descent algorithm has to start somewhere looking for the optimal parameters, and the random set of parameters is one place to start.
So, in a way, we do start with a line, though we don’t draw it. Also, the algorithm itself is not exactly the one presented, of course. The instructor was probably trying to explain it without the notion of a gradient, and it’s tough. So, I’d give him a pass on a sloppy attempt.
