I'm started to read today Shafarevich's book Algebraic Geometry: I could not understand the main idea of the polar line if the point is inside circle. The quote from the book is here:
Here is the most elementary example of this nature. If P is a point outside a circle C then there are two tangent lines to C through P. The line joining their points of contact is called the polar line of P with respect to C. All these constructions can be expressed in terms of algebraic relations between the coordinates of P and the equation of C. Hence they are also applicable to the case that P lies inside C. Of course, the points of tangency of the lines now have complex coordinates, and can’t be seen in the picture. But since the original data was real, the set of points obtained (that is, the two points of tangency) should be invariant on replacing all the numbers by their complex conjugates; that is, the two points of tangency are complex conjugates. Hence the line L joining them is real. This line is also called the polar line of P with respect to C. It is also easy to give a purely real definition of it: it is the locus of points outside the circle whose polar line passes through P.
My questions are:
why the Line L in complex case is Real? What does it mean real in this case?
What does it mean the original data were real and set of points obtained should be invariant on replacing all all the numbers by their complex conjugates?
Thank you in advance,
Could someone to make it clear or tell me where I can read more about it (book, lint etc.)?