The Envelope of Lines, the Astroide
>eq &= y/a+x/sqrt(1-a^2)=1
y x
- + ------------ = 1
a 2
sqrt(1 - a )
>&solve(eq,y); function y(x,a)&=rhs(%[1])
2
a sqrt(1 - a ) - a x
--------------------
2
sqrt(1 - a )
>&assume(x>0); &solve(diff(y(x,a),a)=0,a), function f(x)&=factor(at(y(x,a),%[6]))
2/3 2/3
sqrt(sqrt(3) I x + x + 2)
[a = - -------------------------------,
sqrt(2)
2/3 2/3
sqrt(sqrt(3) I x + x + 2)
a = -------------------------------,
sqrt(2)
2/3 2/3
sqrt(- sqrt(3) I x + x + 2)
a = - ---------------------------------,
sqrt(2)
2/3 2/3
sqrt(- sqrt(3) I x + x + 2) 2/3
a = ---------------------------------, a = - sqrt(1 - x ),
sqrt(2)
2/3
a = sqrt(1 - x )]
2/3 1/3
sqrt(1 - x ) (x - x )
- -------------------------
1/3
x

>&assume(cos(t)>0); &trigsimp(f(sin(t)^3))
3
cos (t)
>a=(0.1:0.05:0.9)'; plot2d("y(x,a)",a=0,b=1,c=0,d=1); insimg;

>function f(x) ...
if x~=0 then return 1;
else if x~=1 then return 0;
else return &:f(x)
endfunction
>plot2d("f",add=1,color=2,thickness=2); insimg;

>eq &= y/a+x/(1-a)=1
y x
- + ----- = 1
a 1 - a
>&solve(eq,y); function y(x,a)&=rhs(%[1])
2
a x + a - a
------------
a - 1
>&assume(x>0); &solve(diff(y(x,a),a)=0,a), function f(x)&=factor(at(y(x,a),%[1]))
[a = 1 - sqrt(x), a = sqrt(x) + 1]
3/2
x - 2 x + sqrt(x)
--------------------
sqrt(x)
>a=(0.1:0.05:0.9)'; plot2d("y(x,a)",a=0,b=1,c=0,d=1);
>function f(x) ...
if x~=0 then return 1;
else if x~=1 then return 0;
else return &:f(x)
endfunction
>plot2d("f",add=1,color=2,thickness=2); insimg;

>eq &= y/a+a*x=1
y
- + a x = 1
a
>&solve(eq,y); function y(x,a)&=rhs(%[1])
2
a - a x
>&assume(x>0); &solve(diff(y(x,a),a)=0,a), function f(x)&=factor(at(y(x,a),%[1]))
1
[a = ---]
2 x
1
---
4 x
>a=exp(-4:0.2:4)'; plot2d("y(x,a)",a=0,b=1,c=0,d=1); ...
>plot2d("1/(4*x)",add=1,color=2,thickness=2); insimg;

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