Solve[{x^2/a^2 - y^2/b^2 == 1, {x + c, y} . {x - c, y} == 0, a > 0,
b > 0, c > 0}, {x, y}]
The software keeps prompting for running, but cannot calculate the result. Why?
$Version
(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)
Clear["Global`*"]
Assuming[{a > 0, b > 0, c > 0},
Solve[{x^2/a^2 - y^2/b^2 == 1, {x + c, y} . {x - c, y} == 0}, {x, y}] //
FullSimplify]
If you also restrict x and y to reals,
Assuming[{a > 0, b > 0, c > 0, Element[{x, y}, Reals]},
Solve[{x^2/a^2 - y^2/b^2 == 1, {x + c, y} . {x - c, y} == 0}, {x, y}] //
FullSimplify]
You can help Mathmatica solve the equations as follows:
ClearAll[a, b, c, x, y, X, Y, X2, Y2];
x = a*X; y = b*Y;
equs = Simplify[{x^2/a^2 - y^2/b^2 == 1, {x+c, y}.{x-c, y} == 0}];
Solve[ equs/. {X^2 -> X2, Y^2 -> Y2}, {X2, Y2}] //Simplify //InputForm
(* {{X2 -> (b^2 + c^2)/(a^2 + b^2), Y2 -> (-a^2 + c^2)/(a^2 + b^2)}} *)
where X2 == (x/a)^2, Y2 == (y/b)^2
.
You asked
The software keeps prompting for running, but cannot calculate the result. Why?
I can not answer that question because it involves detailed knowledge
of the actual software version of Mathematica and algorithms used by
Mathematica. However, if you just want to get a useful solution to
your equations, then I have given that to you using simple replacements
before and after the Solve[]
.
sol = Solve[{x^2/a^2 - y^2/b^2 == 1, {x + c, y} . {x - c, y} == 0, c > a > 0, c > b > 0}, {x, y}, Reals]
$\endgroup$Timing[ByteCount[ sol = ToRadicals[ Reduce[{x^2/a^2 - y^2/b^2 == 1, {x + c, y} . {x - c, y} == 0, c > a > 0, c > b > 0}, {x, y}, Cubics -> True, Quartics -> True]]]]
So what version of Mathematica are you using? $\endgroup$