不定方程式

質問

C1; 12 x^2+7 x y+5 x+y^2+y-9=0
これが双曲線なら漸近線が在るので其れを求めて下さい;
C1∩Z^2を求めて下さい;

C2; 37 x^2-262 x y+6 x+457 y^2-22 y+1=0
これが双曲線なら漸近線が在るので其れを求めて下さい;
C2∩Z^2を求めて下さい;

回答No.1

info33

前半:
C1; 12 x^2+7 x y+5 x+y^2+y-9=0
y=(-7x-1+sqrt((x-3)^2+28)/2
y=(-7x-1-sqrt((x-3)^2+28)/2
双曲線なので漸近線:
y=(-7x-1+(x-3))/2, y= -3x -2 ...(Ans1.)
y=(-7x-1-(x-3))/2, y= -4x -1 ....(Ans2.)

>C1∩Z^2
漸近線交点座標 (3,-11)
C1; 12 (x-3)^2+7 (x-3)(y+11)+5(x-3)+(y+11)^2+(y+11)-9=0
12(x-3)^2+7(x-3)(y+11)+(y+11)^2=7
(3(x-3)+y+11)(4(x-3)+y+11)=7
(3x+y+2)(4x+y-1)=1*7
{3x+y+2=1,4x+y-1=7} or {3x+y+2=-1,4x+y-1=-7} or
{3x+y+2=7,4x+y-1=1} or {3x+y+2=-7,4x+y-1=-1}
(x,y)=(-3,6),(-3,14),(9,-28),(9,-36) ...(Ans.)

後半:
C2; 37 x^2-262 x y+6 x+457 y^2-22 y+1=0
双曲線.
漸近線:y=(3(1316sqrt(7)-131)x+33-5sqrt(7))/1371,
y=(3(6sqrt(7)+131)x+33+5sqrt(7))/1371

>C2∩Z^2
漸近線交点座標:(x,y)=(-5/18,-1/18)
......