#3414 quadratic equatin not solved

[Jack Samhain]

this gives the correct answer:

solve([x²+y²=r², (x-a)²+(y)²=p²],[x,y]);
[[x=(r^2-p^2+a^2)/(2*a),y=-sqrt(-r^4+2*p^2*r^2+2*a^2*r^2-p^4+2*a^2*p^2-a^4)/(2*a)],[x=(r^2-p^2+a^2)/(2*a),y=sqrt(-r^4+2*p^2*r^2+2*a^2*r^2-p^4+2*a^2*p^2-a^4)/(2*a)]]

but this give an empty solution:

solve([x²+y²=r², (x-a)²+(y-b)²=p²],[x,y]);
[]

[Aleksas]

(%i2) eq1:x^2+y^2=r^2$
eq2:(x-a)^2+(y-b)^2=p^2$
(%i3) sol:solve([eq1,eq1-eq2],[x,y]);
(sol)
[[x=-(bsqrt(-r^4+(2p^2+2b^2+2a^2)r^2-p^4+(2b^2+2a^2)p^2-b^4-2a^2b^2-a^4)-ar^2+ap^2-ab^2-a^3)/(2b^2+2a^2),y=(asqrt(-r^4+(2p^2+2b^2+2a^2)r^2-p^4+(2b^2+2a^2)p^2-b^4-2a^2b^2-a^4)+br^2-bp^2+b^3+a^2b)/(2b^2+2a^2)],[x=(bsqrt(-r^4+(2p^2+2b^2+2a^2)r^2-p^4+(2b^2+2a^2)p^2-b^4-2a^2b^2-a^4)+ar^2-ap^2+ab^2+a^3)/(2b^2+2a^2),y=-(asqrt(-r^4+(2p^2+2b^2+2a^2)r^2-p^4+(2b^2+2a^2)p^2-b^4-2a^2b^2-a^4)-br^2+bp^2-b^3-a^2b)/(2b^2+2a^2)]]
Test:
(%i4) subst(sol[1],[eq1,eq2]),radcan;
(%o4) [r^2=r^2,p^2=p^2]
(%i5) subst(sol[2],[eq1,eq2]),radcan;
(%o5) [r^2=r^2,p^2=p^2]

2018-03-23 10:29 GMT+02:00 Jack Samhain jacksamhain@users.sourceforge.net:

---

• [bugs:#3414] https://sourceforge.net/p/maxima/bugs/3414/ quadratic
  equatin not solved *

Status: open
Group: None
Created: Fri Mar 23, 2018 08:29 AM UTC by Jack Samhain
Last Updated: Fri Mar 23, 2018 08:29 AM UTC
Owner: nobody

this gives the correct answer:

solve([x²+y²=r², (x-a)²+(y)²=p²],[x,y]);
[[x=(r^2-p^2+a^2)/(2a),y=-sqrt(-r^4+2p^2r^2+2a^2r^2-p^4+2a^2p^2-a^4)/(2a)],[x=(r^2-p^2+a^2)/(2a),y=sqrt(-r^4+2p^2r^2+2a^2r^2-p^4+2a^2p^2-a^4)/(2a)]]

but this give an empty solution:

solve([x²+y²=r², (x-a)²+(y-b)²=p²],[x,y]);
[]

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[Jack Samhain]

Ok, found it: I have here these versions from the debian repository:
maxima 5.38.1-8+b1
wxmaxima 16.04.2-1

"maxima" gives the correct result, just like you quoted.
"wxmaxima" gives the incorrect result.

So this is a bug in wxmaxima, not maxima.

[Jack Samhain]

Did not work out so well. Afterupgrading ti the latest supported version I get reliably this result:

(%i9) eq1:x²+y²=r²$
eq2:(x-a)²+(y+b)²=d²$
solve([eq1,eq2],[x,y]);
(%o9) []

This happens on maxima 5.38.1-8+b1 (devuan) and 5.41.0 (FreeBSD).

[David Billinghurst]

I'll take this one.

[David Billinghurst]

solve calls algsys to solve systems of polynomial equations. The problem here is (probably) too general for algsys. If some parameters are given values then solutions are found. I will investigate a little further but it is probably a wont-fix.

What I recommend for this type of problem is to use geometric insight to eliminate some degrees of freedom. The problem as posed is the intersection of a circle centre (0.0) radius r and a circle centre (a,-b) radius d. By rotating and scaling you can transform the problem to the system of equations at (%i8) which is solvable.

(%i1) display2d:false;
(%o1) false
(%i2) eq1:x^2+y^2=r^2$
(%i3) eq2:(x-a)^2+(y+b)^2=d^2$
(%i4) solve([eq1,eq2],[x,y]);
(%o4) []
(%i5) solve([eq1,eq2],[x,y]),a=1,b=1,r=2,d=3;
(%o5) [[x = -(sqrt(23)+3)/4,y = -(sqrt(23)-3)/4],
       [x = (sqrt(23)-3)/4,y = (sqrt(23)+3)/4]]
(%i6) solve([eq1,eq2],[x,y]),a=1,b=1,r=2;
(%o6) []
(%i7) solve([eq1,eq2],[x,y]),a=1,b=0,r=2;
(%o7) [[x = -(d^2-5)/2,y = -sqrt((-d^4)+10*d^2-9)/2],
       [x = -(d^2-5)/2,y = sqrt((-d^4)+10*d^2-9)/2]]
 (%i8) solve([eq1,eq2],[x,y]),a=1,b=0;
(%o8) [[x = (r^2-d^2+1)/2,
        y = -sqrt((-r^4)+(2*d^2+2)*r^2-d^4+2*d^2-1)/2],
       [x = (r^2-d^2+1)/2,
        y = sqrt((-r^4)+(2*d^2+2)*r^2-d^4+2*d^2-1)/2]]

[Jack Samhain]

You are right, one can take the solver by it's hand and make it solve the equations, but that is not the point: The solver fails at this simple problem. How should the user trust the solver to work correctly, when it can not solve a perfectly solvable problem? It's a pitty, but in my case it disqualifies maxima for use in classes.

My old TI-92 solves this equations the correct way :)