This is a long question, and might seem like a repost of my earlier questions, but it isn't, hear me out:
In my book is written:
The equation of the line tangent to the circle $x^2+y^2=r^2$ in the point $P(a,b)$ is $ax+by=r^2$. This is only if $P$ is a point on the circle. However, if $P$ is not a point on the circle, you can find the line which goes through the points of tangency. Here an example of such a line (called poollijn in Dutch, don't know the English name, I'll keeping reffering to it as poollijn ):
Now we have the following question:
Find the equations of the lines from $P(0,6)$ tangent to the circle $x^2+y^2=4x+4$
I have posted a question about this one on the forum before, and I solved it afterwards by simply eliminating $y$. However, I ALSO wanted to know how to solve this problem using the poollijn. Sadly, I haven't yet. When I went to check the correction sheet, I saw that I made a mistake right at the beginning, these are the first steps:
$xa+by=4x+4$
$ x.0+y.6 = 2x + 2.0 +4 $
$6y = 2x+4$
$y= \dfrac{1}{3}x + \dfrac{2}{3}$ (poollijn)
What I don't understand, is the second step. Why does the RHS suddenly change from $4x+4$ to $2x+2.0+4$? It must be right too, since the final answers you get by using this method (first finding the poollijn, then the intersection with the circle, and then you have 2 points per line, so you can make the equations) are correct. So what am I seeing incorrectly. Why the change from $4x+4$ to $2x+4$