show that this equation 'ellipse'
peter.vlasschaert@gmail.com,28/03/2017
ref:
http://staff.argyll.epsb.ca/jreed/math30p/conics/general.htm

kill(all)$
load(eigen)$

given : parametric  x=x(t)
                    y=y(t),k=constant

p1:X=k*cos(t);
p2:Y=-(k*sin(t)+k*cos(t))/2;

p3:linsolve([p1, p2], [cos(t),sin(t)]);

p4:p3[1];
p5:p3[2];

p6:(p3[1])^2;
p7:(p3[2])^2;

p8:(p6+p7);

geometry: identity '(cos(theta))^2+(sin(theta))^2 = 1'

p9:trigrat(sin(t)^2+cos(t)^2=(2*Y+X)^2/k^2+X^2/k^2)*k^2;

ref : A*x^2+B*X*Y+C*y^2+D*x+E*x+F = 0 (1)

p10:rhs(p9)-lhs(p9)=0;

p11:A=coeff(part(p10,1),X^2);
p111:ratsubst(v,X*Y, part(p10,1)=0)$
p12:B=coeff(part(p111,1),v);
p13:C=coeff(part(p10,1),Y^2);
p14:D=ev(coeff(part(p10,1),X),Y=0);
p15:E=ev(coeff(part(p10,1),Y),X=0);
p16:F=ev(part(p10,1),X=0,Y=0);

X*Y : rotation of the conic section.
-----------------------------------
see:ref = ' det =  B^2-4*A*C ' (2)
-----------------------------------
see ' above (1) '
-----------------------------------
circle   : A=C<>0 , det < 0
ellipse  : A<>C   , det < 0
parabola :        , det = 0
hyperbola:        , det > 0
-----------------------------------
-----------------------------------

p17:det=(part(p12,2))^2-4*(part(p11,2))*(part(p13,2));

programming : Maxima

rem : construction ' simple if '

if part(p17,2)<0 and part(p11,2)=part(p13,2) then 
 disp("circle")
else 
 disp("ellipse")$
if part(p17,2)> 0 then
 disp("hyperbola")$
if part(p17,2)= 0 then
 disp("parabola")$

# = not equal to 
other :>,=,<,>=,<=

if part(p12,2) # 0 then
 disp ("rotated conic ")$

rem : construction: if 'with elseif'
      if ' condition ' then
         output
      elseif ' condition ' then output
      elseif ' condition ' then output
      else output

s:3;

if s < 0 then
 "minus" 
elseif 
s < 5 then "small" 
elseif 
s < 10 then "medium" 
else
 "large";

appendix:  statement 'trigrat'