# Question:How to write the equation of the planes tangent to the sphere at the points in this list?

## Question:How to write the equation of the planes tangent to the sphere at the points in this list?

Maple

list:= [[-12, 3, 5], [-11, -2, 5], [-11, -1, 2], [-11, -1, 8], [-11, 0,
1], [-11, 0, 9], [-11, 3, 0], [-11, 3, 10], [-11, 6, 1], [-11, 6,
9], [-11, 7, 2], [-11, 7, 8], [-11, 8, 5], [-4, -9, 5], [-4,
3, -7], [-4, 3, 17], [-4, 15, 5], [-3, -9, 2], [-3, -9, 8], [-3,
0, -7], [-3, 0, 17], [-3, 6, -7], [-3, 6, 17], [-3, 15, 2], [-3, 15,
8], [-2, -9, 1], [-2, -9, 9], [-2, -1, -7], [-2, -1, 17], [-2,
7, -7], [-2, 7, 17], [-2, 15, 1], [-2, 15, 9], [1, -10, 5], [1, -9,
0], [1, -9, 10], [1, -2, -7], [1, -2, 17], [1, 3, -8], [1, 3,
18], [1, 8, -7], [1, 8, 17], [1, 15, 0], [1, 15, 10], [1, 16,
5], [4, -9, 1], [4, -9, 9], [4, -1, -7], [4, -1, 17], [4,
7, -7], [4, 7, 17], [4, 15, 1], [4, 15, 9], [5, -9, 2], [5, -9,
8], [5, 0, -7], [5, 0, 17], [5, 6, -7], [5, 6, 17], [5, 15, 2], [5,
15, 8], [6, -9, 5], [6, 3, -7], [6, 3, 17], [6, 15, 5], [13, -2,
5], [13, -1, 2], [13, -1, 8], [13, 0, 1], [13, 0, 9], [13, 3,
0], [13, 3, 10], [13, 6, 1], [13, 6, 9], [13, 7, 2], [13, 7,
8], [13, 8, 5], [14, 3, 5]];
These points are lies on the sphere (x - 1)^2 + (y - 3)^2 + (z - 5)^2 = 13^2. How to write the equation of the planes tangent to the sphere at the points in the list? This is some equations.
I tried

restart;
with(geom3d);
L := [[-12, 3, 5], [-11, -2, 5], [-11, -1, 2], [-11, -1, 8], [-11, 0, 1], [-11, 0, 9], [-11, 3, 0], [-11, 3, 10], [-11, 6, 1], [-11, 6, 9], [-11, 7, 2], [-11, 7, 8], [-11, 8, 5], [-4, -9, 5], [-4, 3, -7], [-4, 3, 17], [-4, 15, 5], [-3, -9, 2], [-3, -9, 8], [-3, 0, -7], [-3, 0, 17], [-3, 6, -7], [-3, 6, 17], [-3, 15, 2], [-3, 15, 8], [-2, -9, 1], [-2, -9, 9], [-2, -1, -7], [-2, -1, 17], [-2, 7, -7], [-2, 7, 17], [-2, 15, 1], [-2, 15, 9], [1, -10, 5], [1, -9, 0], [1, -9, 10], [1, -2, -7], [1, -2, 17], [1, 3, -8], [1, 3, 18], [1, 8, -7], [1, 8, 17], [1, 15, 0], [1, 15, 10], [1, 16, 5], [4, -9, 1], [4, -9, 9], [4, -1, -7], [4, -1, 17], [4, 7, -7], [4, 7, 17], [4, 15, 1], [4, 15, 9], [5, -9, 2], [5, -9, 8], [5, 0, -7], [5, 0, 17], [5, 6, -7], [5, 6, 17], [5, 15, 2], [5, 15, 8], [6, -9, 5], [6, 3, -7], [6, 3, 17], [6, 15, 5], [13, -2, 5], [13, -1, 2], [13, -1, 8], [13, 0, 1], [13, 0, 9], [13, 3, 0], [13, 3, 10], [13, 6, 1], [13, 6, 9], [13, 7, 2], [13, 7, 8], [13, 8, 5], [14, 3, 5]];
eqS := Equation(sphere(S, (x - 1)^2 + (y - 3)^2 + (z - 5)^2 - 169 = 0, [x, y, z], 'centername' = T));
k := [seq](sort(Equation(TangentPlane(P, S, point(A, pt[])), [x, y, z])), pt in L);
seq([L[i], k[i]], i = 1 .. nops(L));

I get

With each equation has the form a*x  + b*y + c*x + d=0, how can I get in the form igcd(a,b,c,d) = 1 and a >0. If a = 0, then b>0; If b = 0, then c>0.

PS. I use this expr := sort(primpart(3*x + 6*y + 12));
expr*signum(lcoeff(expr)) = 0;

But I do not know use it in the seq k

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