Ytukyg

I am trying to write equation of the line passing through two points pA={1, -3} and pB={-33, -1} in the form x+17 y+50=0. I tried

{pA, pB} = {{1, -3}, {-33, -1}};

u = pB - pA;

m = {x, y};

v = m - pA;

d = Det[{u, v}];

w = {Coefficient[d, x], Coefficient[d, y]};

k = GCD[Coefficient[d, x], Coefficient[d, y]];

If[w[[1]] != 0, n = Sign[w[[1]]] w/k, 

  If[w[[2]] != 0, n = Sign[w[[2]]] w/k]];

TraditionalForm[Expand[n.v]] == 0

I got

x+17 y+50==0

Is there another way to write it?

Simplify[y - InterpolatingPolynomial[{pA, pB}, x] == 0]

50 + x + 17 y == 0

Also

Simplify[y - a x - b == 0 /. First@Solve[a # + b == #2 & @@@ {pA, pB}, {a, b}]]

50 + x + 17 y == 0

And

Simplify @ Rationalize[y - Fit[{pA, pB}, {x, 1}, x] == 0]

50 + x + 17 y == 0

You may do as follows. Let us look for the equation in the form ax+by==1, where the parameters a and b are to be found.This will substitute the coordinates of the points pA and pB into this equations, thus, forming two equations with respect to a and b and solves the system:

eq = a*#[[1]] + b*#[[2]] == 1 &;

eq1=eq /@ {{1, -3}, {-33, -1}}



(*   {a - 3 b == 1, -33 a - b == 1}  *)

This will substitute the solution into the linear equation already in coordinates x and y:

eq[{x, y}] /. sol



(* -(x/50) - (17 y)/50 == 1 *)

This will plot the solution:

Show[{

  ContourPlot[-(x/50) - (17 y)/50 == 1, {x, -34, 2}, {y, -4, 0}],

  Graphics[{Red, PointSize[0.015], Point[#] & /@ {{1, -3}, {-33, -1}}}]

  }]

yielding the following plot:

The original points are shown in red.

This is one of several possible ways.

Have fun!

The equation of a planar line going through two points $ P_1(x_1,y_1) $ and $ P_2(x_2,y_2) $ (e.g. cf this for reference) is

$$
begin{vmatrix}
x & y\
x_2-x_1 & y_2-y_1
end{vmatrix}
=
begin{vmatrix}
x_1 & y_1\
x_2 & y_2
end{vmatrix}.
$$

So there is the piece of codes below:

Clear[eq, pts]

eq = Simplify[Det[{{x, y}, -Subtract @@ #}] == Det[#]] &;

pts = {{1, -3}, {-33, -1}};

eq[pts]

50 + x + 17 y == 0

Or

eq2 = Simplify[Det[{-1, 1} Differences[Prepend[#, {x, y}]]] == 0] &;

With RegionMember:

Simplify[RegionMember[InfiniteLine[{{1, -3}, {-33, -1}}], {x, y}], 

 Element[x | y, Reals]]

50 + x + 17 y == 0

Knowing that the coefficients are components of a vector perpendicular to the difference of the two points, I think the most convenient command to obtain the equation is

perp = Cross[pB - pA];

perp.{x, y} == perp.pA // Simplify

50 + x + 17 y == 0

The last step before Simplify is

-2 x - 34 y == 100

so you can see that the simplification brought all terms to one side, factored out the greatest common divisor and fixed the signs.

To Mathematica the sums "50 + x + 17y" and "x + 17y + 50" are exactly the same expression, but if you want to order linear terms before constants in the displayed form, you may consider using TraditionalForm (with the added benefit of using a "normal" equality sign while remaining copy-and-pastable):

% // TraditionalForm

$x+17 y+50=0$