In[]:=
ClearAll
Out[]=
ClearAll
In[]:=
eqns={x^2/a^2+y^2/b^21,ykx+m};​​polyex=Apply[Subtract,eqns,{1}];​​polys=Numerator[Together[Apply[Subtract,eqns,{1}]]];​​xpoly=Collect[Resultant[polys[[1]],polys[[2]],y],x]​​discx=Factor[Discriminant[xpoly,x]](*discriminant*)​​frist=Solve[eqns,{x,y}]//FullSimplify;​​{{x1,y1},{x2,y2}}={x,y}/.frist;​​second={x1+x2,x1x2,y1+y2,y1y2}//FullSimplify​​thrid={x1x2+y1y2,x1y2+x2y1}//FullSimplify​​slope=-CoefficientList[polyex[[2]],x][[2]];(*k*)​​intercept=-CoefficientList[CoefficientList[polyex[[2]],y][[1]],x][[1]];(*m*)​​Chordlength=FullSimplify[Sqrt[1+slope^2]Sqrt[(x1+x2)^2-4x1x2]](*AbsAB*)
Out[]=
-
2
a
2
b
+
2
a
2
m
+2
2
a
kmx+(
2
b
+
2
a
2
k
)
2
x
Out[]=
4
2
a
2
b
(
2
b
+
2
a
2
k
-
2
m
)
Out[]=
-
2
2
a
km
2
b
+
2
a
2
k
,
2
a
(-
2
b
+
2
m
)
2
b
+
2
a
2
k
,
2
2
b
m
2
b
+
2
a
2
k
,
2
b
(-
2
a
2
k
+
2
m
)
2
b
+
2
a
2
k

Out[]=

-
2
a
2
b
(1+
2
k
)+(
2
a
+
2
b
)
2
m
2
b
+
2
a
2
k
,-
2
2
a
2
b
k
2
b
+
2
a
2
k

Out[]=
2
1+
2
k
2
a
2
b
(
2
b
+
2
a
2
k
-
2
m
)
2
(
2
b
+
2
a
2
k
)
In[]:=
ClearAll
Out[]=
ClearAll
In[]:=
eqns={x^2/a^2+y^2/b^21,xty+n};​​polyex=Apply[Subtract,eqns,{1}];​​polys=Numerator[Together[Apply[Subtract,eqns,{1}]]];​​xpoly=Collect[Resultant[polys[[1]],polys[[2]],x],y]​​discx=Factor[Discriminant[xpoly,y]](*discriminant*)​​frist=Solve[eqns,{x,y}]//FullSimplify;​​{{x1,y1},{x2,y2}}={x,y}/.frist;​​second={x1+x2,x1x2,y1+y2,y1y2}//FullSimplify​​thrid={x1x2+y1y2,x1y2+x2y1}//FullSimplify​​slope=-CoefficientList[polyex[[2]],y][[2]];(*k*)​​intercept=-CoefficientList[CoefficientList[polyex[[2]],y][[1]],x][[1]];(*m*)​​Chordlength=FullSimplify[Sqrt[1+slope^2]Sqrt[(y1+y2)^2-4y1y2]](*AbsAB*)
Out[]=
-
2
a
2
b
+
2
b
2
n
+2
2
b
nty+(
2
a
+
2
b
2
t
)
2
y
Out[]=
4
2
a
2
b
(
2
a
-
2
n
+
2
b
2
t
)
Out[]=

2
2
a
n
2
a
+
2
b
2
t
,
2
a
(n-bt)(n+bt)
2
a
+
2
b
2
t
,-
2
2
b
nt
2
a
+
2
b
2
t
,
2
b
(-
2
a
+
2
n
)
2
a
+
2
b
2
t

Out[]=

2
a
2
n
+
2
b
2
n
-
2
a
2
b
(1+
2
t
)
2
a
+
2
b
2
t
,-
2
2
a
2
b
t
2
a
+
2
b
2
t

Out[]=
2
1+
2
t
2
a
2
b
(
2
a
-
2
n
+
2
b
2
t
)
2
(
2
a
+
2
b
2
t
)
In[]:=
ClearAll
Out[]=
ClearAll
In[]:=
eqns={x^2/a^2+y^2/b^21,yk(x-x0)+y0};​​polyex=Apply[Subtract,eqns,{1}];​​polys=Numerator[Together[Apply[Subtract,eqns,{1}]]];​​xpoly=Collect[Resultant[polys[[1]],polys[[2]],y],x]​​discx=Factor[Discriminant[xpoly,x]](*discriminant*)​​frist=Solve[eqns,{x,y}]//FullSimplify;​​{{x1,y1},{x2,y2}}={x,y}/.frist;​​second={x1+x2,x1x2,y1+y2,y1y2}//FullSimplify​​thrid={x1x2+y1y2,x1y2+x2y1}//FullSimplify​​slope=-CoefficientList[polyex[[2]],x][[2]];(*k*)​​intercept=-CoefficientList[CoefficientList[polyex[[2]],y][[1]],x][[1]];(*m*)​​Chordlength=FullSimplify[Sqrt[1+slope^2]Sqrt[(x1+x2)^2-4x1x2]](*AbsAB*)
Out[]=
-
2
a
2
b
+(
2
b
+
2
a
2
k
)
2
x
+
2
a
2
k
2
x0
-2
2
a
kx0y0+
2
a
2
y0
+x(-2
2
a
2
k
x0+2
2
a
ky0)
Out[]=
4
2
a
2
b
(
2
b
+
2
a
2
k
-
2
k
2
x0
+2kx0y0-
2
y0
)
Out[]=

2
2
a
k(kx0-y0)
2
b
+
2
a
2
k
,-
2
a
(
2
b
-
2
(-kx0+y0)
)
2
b
+
2
a
2
k
,
2
2
b
(-kx0+y0)
2
b
+
2
a
2
k
,-
2
b
(
2
a
2
k
-
2
(-kx0+y0)
)
2
b
+
2
a
2
k

Out[]=

2
b
2
(-kx0+y0)
+
2
a
(-
2
b
(1+
2
k
)+
2
(-kx0+y0)
)
2
b
+
2
a
2
k
,-
2
2
a
2
b
k
2
b
+
2
a
2
k

Out[]=
2
1+
2
k
2
a
2
b
(
2
b
+
2
a
2
k
-
2
(-kx0+y0)
)
2
(
2
b
+
2
a
2
k
)
In[]:=
Collect[
2
b
+
2
a
2
k
-
2
k
2
x0
+2kx0y0-
2
y0
,k]
Out[]=
2
b
+
2
k
(
2
a
-
2
x0
)+2kx0y0-
2
y0
In[]:=
eqns={x^2/4+y^2/31,yk(x-x0)+y0};​​polyex=Apply[Subtract,eqns,{1}];​​polys=Numerator[Together[Apply[Subtract,eqns,{1}]]];​​xpoly=Collect[Resultant[polys[[1]],polys[[2]],y],x]​​discx=Factor[Discriminant[xpoly,x]](*discriminant*)​​frist=Solve[eqns,{x,y}]//FullSimplify;​​{{x1,y1},{x2,y2}}={x,y}/.frist;​​second={x1+x2,x1x2,y1+y2,y1y2}//FullSimplify​​thrid={x1x2+y1y2,x1y2+x2y1}//FullSimplify​​slope=-CoefficientList[polyex[[2]],x][[2]];(*k*)​​intercept=-CoefficientList[CoefficientList[polyex[[2]],y][[1]],x][[1]];(*m*)​​Chordlength=FullSimplify[Sqrt[1+slope^2]Sqrt[(x1+x2)^2-4x1x2]](*AbsAB*)
Out[]=
-12+(3+4
2
k
)
2
x
+4
2
k
2
x0
-8kx0y0+4
2
y0
+x(-8
2
k
x0+8ky0)
Out[]=
-48(-3-4
2
k
+
2
k
2
x0
-2kx0y0+
2
y0
)
Out[]=

8k(kx0-y0)
3+4
2
k
,
4(-3+
2
(-kx0+y0)
)
3+4
2
k
,
-6kx0+6y0
3+4
2
k
,
3(
2
k
(-4+
2
x0
)-2kx0y0+
2
y0
)
3+4
2
k

Out[]=

-12+
2
k
(-12+7
2
x0
)-14kx0y0+7
2
y0
3+4
2
k
,-
24k
3+4
2
k

Out[]=
4
3
1+
2
k
-
-3+
2
k
(-4+
2
x0
)-2kx0y0+
2
y0
2
(3+4
2
k
)
In[]:=
Collect[-12+(3+4
2
k
)
2
x
+4
2
k
2
x0
-8kx0y0+4
2
y0
+x(-8
2
k
x0+8ky0),x]//FullSimplify
Out[]=
(3+4
2
k
)
2
x
+8kx(-kx0+y0)+4(-3+
2
(-kx0+y0)
)