<s>[INST] Problem regarding the distance between a circle and a parabola The question is this, copy pasted here:
Consider the circle $C$ whose equation is
$$(x - 2) ^ 2 + (y - 8) ^ 2 = 1$$
and the parabola $P$ with the equation
$$y^ 2 =4x.$$
I have no idea how to model the question, I tried showing that the $x-$coordinates of the two points have to be the same but that seems to be wrong too? Can anyone help?
 [/INST] Let $(x,y) $ be on the parbola, then
$ y^2 = 4 x$
The normal vector is $[ 4, -2 y] $ , while the vector from $(2, 8)$ to $(x,y)$ is $[ x - 2, y - 8 ]$, and we want these two vectors to be aligned; so we can use the determinant and set it equal to zero, namely,
$4 (y-8) + 2 y (x - 2) = 0 $
We have to solve the above equation together with $ y^2 = 4 x $
Hence,
$ 4 ( y - 8) + 2 y (y^2/4 - 2) = 0 $
From which
$ -64 + y^3 = 0$
Whose solution is $ y = 4 $, and therefore, $ x = y^2 /4 = 4 $
This means that the shortest distance between the circle and the parabola is
$ d = \sqrt{ (2 - 4)^2 + (8 - 4)^2 } - 1 = \sqrt{20} - 1 = 2 \sqrt{5} - 1 $
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