In[]:=
Collect[SubtractSides[Eliminate[{x^2/a^2-y^2/b^21,ykx+m},y],x],x]​​SubtractSides[%]​​MultiplySides[%,a^2,GenerateConditionsFalse]​​Collect[%,x,Apart]
Out[]=
2
m
+(-1+2km)x-
2
b
-x+
2
b
2
a
-
2
k
2
x
&&a-x≠-x&&b-x≠-x
Out[]=
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&a≠0&&b≠0
Out[]=
2
a
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&
3
a
≠0&&
2
a
b≠0
Out[]=
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0&&
3
a
≠0&&
2
a
b≠0
In[]:=
Collect[SubtractSides[Eliminate[{-x^2/a^2+y^2/b^21,ykx+m},y],x],x]​​SubtractSides[%]​​MultiplySides[%,a^2,GenerateConditionsFalse]​​Collect[%,x,Apart]
Out[]=
2
m
+(-1+2km)x
2
b
-x+
2
b
2
a
-
2
k
2
x
&&a-x≠-x&&b-x≠-x
Out[]=
-
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&a≠0&&b≠0
Out[]=
2
a
-
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&
3
a
≠0&&
2
a
b≠0
Out[]=
-
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0&&
3
a
≠0&&
2
a
b≠0
In[]:=
#[[2]]^2-4#[[1]]#[[3]]&@CoefficientList[-
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
,{x}]​​
Out[]=
4
4
a
2
k
2
m
-4(-
2
b
+
2
a
2
k
)(-
2
a
2
b
+
2
a
2
m
)
In[]:=
Simplify[4
4
a
2
k
2
m
-4(-
2
b
+
2
a
2
k
)(-
2
a
2
b
+
2
a
2
m
)]
Out[]=
4
2
a
2
b
(-
2
b
+
2
a
2
k
+
2
m
)
In[]:=
Simplify[#[[2]]^2-4#[[1]]#[[3]]&@CoefficientList[
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
,{x}]]
Out[]=
4
2
a
2
b
(
2
b
-
2
a
2
k
+
2
m
)
In[]:=
Collect[SubtractSides[Eliminate[{x^2/a^2-y^2/b^21,ykx+m},y],x],x]​​SubtractSides[%]​​MultiplySides[%,a^2,GenerateConditionsFalse]​​Collect[%,x,Apart]
Out[]=
2
m
+(-1+2km)x-
2
b
-x+
2
b
2
a
-
2
k
2
x
&&a-x≠-x&&b-x≠-x
Out[]=
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&a≠0&&b≠0
Out[]=
2
a
2
b
+
2
m
+x+(-1+2km)x-
2
b
2
a
-
2
k
2
x
0&&
3
a
≠0&&
2
a
b≠0
Out[]=
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0&&
3
a
≠0&&
2
a
b≠0
Simplify[#[[2]]^2-4#[[1]]#[[3]]&@CoefficientList[
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
,{x}]]
Out[]=
4
2
a
2
b
(
2
b
-
2
a
2
k
+
2
m
)
In[]:=
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0
Out[]=
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0
In[]:=
Solve[
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(-
2
b
+
2
a
2
k
)
2
x
0,{x},]
Out[]=
x
-
2
a
km
-
2
b
+
2
a
2
k
-
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
if
condition
,x
-
2
a
km
-
2
b
+
2
a
2
k
+
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
if
condition

In[]:=
x1=-
2
a
km
-
2
b
+
2
a
2
k
-
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
;x2=-
2
a
km
-
2
b
+
2
a
2
k
+
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
Out[]=
-
2
a
km
-
2
b
+
2
a
2
k
+
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
In[]:=
x1+x2
Out[]=
-
2
2
a
km
-
2
b
+
2
a
2
k
x1x2
Out[]=
-
2
a
km
-
2
b
+
2
a
2
k
-
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
-
2
a
km
-
2
b
+
2
a
2
k
+
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
In[]:=
Simplify-
2
a
km
-
2
b
+
2
a
2
k
-
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)
-
2
a
km
-
2
b
+
2
a
2
k
+
2
a
4
b
-
4
a
2
b
2
k
+
2
a
2
b
2
m
2
(
2
b
-
2
a
2
k
)

Out[]=
2
a
(
2
b
+
2
m
)
-
2
b
+
2
a
2
k