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Last edited by hbghlyj 2020-12-25 11:02例如,(0,0)是$y^2=(x-1)x^2$的孤立点.
(1,-1)是$\left(x-2y-4\right)\left(\left(x-1\right)^2+\left(y+1\right)^2\right)=0$的孤立点.
(0,0)是$x^{4}+y^{4}=x^{2}y^{2}\left(x^{2}+y^{2}\right)$的孤立点.
(-3,0)是$(x+1)(y^2-x-5)=4$的孤立点.
证明:这个方程等价于$(x+1)(y+1)^2=(x+3)^2$
(1,-1)是$\left(x-2\right)\left(3x^2-5y^2+2x-10y-2\right)=-8$的孤立点
证明:这个方程等价于$5(x-2)(y + 1)^2= (x-1)^2(3x+2)$
$\left(-\frac{10}9,-1\right)$是$\left(x-1\right)\left(3x^2+8x+8+5\left(y+1\right)^2\right)=-\frac{1444}{243}$的孤立点
证明:这个方程等价于$1215(x-1)(y+1)^2=(9x-5)(9x+10)^2$
$\left(\frac 83,-1\right)$是$\left(x-2\right)\left(9x^{2}+15y^{2}+6x+30y-101\right)=-24$的孤立点
证明:这个方程等价于$15\left(x-2\right)\left(y+1\right)^{2}=-\left(x+4\right)\left(3x-8\right)^{2}$
一般地,给出曲线的隐式方程,如何求得所有孤立点? |
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