Intersection of a line and a circle and distance between them

Question

i'm trying to plot the intersection point between a line and a circle but it's not working. How can I do that using the following command?

Manipulate[Show[ParametricPlot[{r Cos[t] + c, r Sin[t] + d}, {t, 0, 2 Pi}, PlotRange -> {-10, 10}],Plot[Tan[t] x + b, {x, -5 Pi, 5 Pi}]], {t, 0, 2 Pi}, {b, -40, 40}, {r, 0, 8}, {c, -10, 10}, {d, -10, 10}]

I also want to measure the shortest distance between that circle and the line. Am I able to do that using the same variables?

Answer 1

Here is an example that demonstrates several functions/features that you might want to use:

Manipulate[
  With[{
    regionCircle = Circle[{centerX, centerY}, radius],
    regionLine = InfiniteLine[{0, intercept}, {1, slope}],
    points = 
      NSolveValues[{(x - centerX)^2 + (y - centerY)^2 == radius^2 && y == slope x + intercept}, {x, y}, Reals]},
    Column[{
      Graphics[{
        Blue, regionCircle,
        Purple, regionLine,
        Red, PointSize[.05], Point[points]},
        PlotRange -> {{-20, 20}, {-20, 20}}],
      Row[{"intersection points: ", points}],
      Row[{"distance to center: ", RegionDistance[regionLine, {centerX, centerY}]}],
      Row[{"distance to circumference: ", Max[0, RegionDistance[regionLine, {centerX, centerY}] - radius]}]}]],
  {{centerX, 0, "center x"}, -10, 10},
  {{centerY, 0, "center y"}, -10, 10},
  {{radius, 5, "radius"}, 1, 10},
  {{intercept, 0, "y intercept"}, -10, 10},
  {{slope, 0, "slope"}, -10, 10}]

Answer 2

Here my answer using region functionality and userdefined distance function:

Manipulate[kreis = Circle[{xc, yc}, rc];

gerade = {xg, yg} + {Cos[tg], Sin[tg]} u;

mini = NMinimize[# . # &[gerade - {xc, yc}], u];(* minimal distance to circle center*)
dist = If[Sqrt[mini[[1]]] <= rc, 0, Sqrt[mini[[1]]] - rc];
pg = gerade /. mini[[2]];(* nearest linepoint*)
linie = InfiniteLine[{gerade /. u -> 0, gerade /. u -> 1} ];
pts = RegionIntersection[kreis, linie];

Graphics[{kreis, linie, Red, pts, Blue, Point[pg]}, 
PlotLabel -> "RegionDistance " <> ToString[dist],ImageSize->200], {{rc, 1}, 0,8}, {{xc, 0}, -10, 10}, {{yc, 0}, -10, 10}, {{xg, 2}, -10, 10}, {{yg, 1}, -10, 10}, {{tg, 1}, -1,1}]