Let  a  and  b  be the major and minor semiaxes of the ellipse, c - focal distance. From the condition  b=с  and the equality  a^2=b^2+c^2 , we easily get that  a=c*sqrt(2)  and the eccentricity  e=1/sqrt(2) .  By the condition b = c  the ellipse is determined up to similarity, the alpha angle does not change, therefore, in the calculations, we can assume that  c=1 .
Below  r1  and  r2  are the distances from a point  M(x,y)  on the ellipse to the ellipse foci.

Download max_angle.mw

Use  op(eqs)  or  eqs[ ]  for this.

restart;
S:={a*x,b*x, c*(x+y)};
S1:=op~(S);
S1 minus {a,b,c};

restart;
eq:=sqrt(y)=tanh(x);
eq^2;

Edit.

Download PMSM_eq_new1.mw

In each specific example, the shortest way is probably to use the  subsop  command:

restart;	
sol:={v[1]=t,v[2]=3/2*t,v[3]=v[3]}:
subsop(3=(),sol);

R:=1.33*10^(-9)*C/m^2;
evalf[3](op(1,R)*``(`*`(op(2..3,R))));

restart;
expr:=exp(x)^3*sin(x)+3/(exp(x)^n):
A:=exp(t::anything)^n::anything=exp(t*n):
applyrule([1/A,A], expr);

Edit.

restart;
a:= 456;
L:=convert(a,base,10);
Array([seq(L[i],i=-1..-nops(L),-1)]);
# Or shorter
Array(ListTools:-Reverse(L));

Edit.

Use  LinearAlgebra:-Modular  subpackage to work with matrices on a specific finite field.

To generate all such matrices, read the thread  https://mapleprimes.com/questions/211872-Generate-Invertible-4x4-Matrices-Over-F2-

restart;
x, y := r*cos(t), r*sin(t):
plot3d([[x, y, r], [x, y, 0]], r= 0..1, t= 0..Pi, scaling= constrained);

Your equation contains several parameters in addition to variables  x  and  y . For Maple to accept this equation as an ellipse equation, conditions on the parameters are necessary. To get these conditions, we first select complete squares:

restart;
with(geometry):
eqell := expand((x+(1/2)*R1-(1/2)*R)^2/a^2+y^2/b^2-1);
A:=Student:-Precalculus:-CompleteSquare(eqell,x,y);
B, C := selectremove(t->has(t,x)or has(t,y), A);  
Eq:=subsindets(B,`^`,t->applyop(simplify,1,t))=simplify(-C); # Simplified ellipse equation
geometry:-ellipse(ell, Eq, [x, y]) assuming a>0, b>0, a>b; 
detail(ell);

Edit.

Download Ellipse1.mw

Use the appropriate options. An example:

restart;
plot([sin(x),cos(x)], x=0..2*Pi, color=[blue,red], style=point, symbol=[cross,diagonalcross], symbolsize=12, numpoints=50, adaptive=false, size=[800,400], scaling=constrained);

See help on  ?plot,options
If you want to have a solid line as well, and not just some symbols, then use  style=pointline  option.

Remember that  e  is just a Maple symbol (not Euler's number e), and the exponential function  e^x  should be coded as  exp(x) .
First, we plot this function to ensure the existence and uniqueness of the root.
 

Download bisect.mw

acer's way makes sense for a single application. But if this needs to be done several times, then a more significant gain (for arbitrary matrices) is given by applying the procedure:
 

Download Columns.mw