Section 6.4

Affine Types of Second-Order Curves

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Problem 557

Determine the affine type of the following curves:

?
(1)

x2+2xy+y2+y=0x^{2}+2 x y+y^{2}+y = 0

(2)

x2+2xy+y2+x+y=0x^{2}+2 x y+y^{2}+x+y = 0;

(3)

(x+2y)2−3y2=1(x+2 y)^{2}-3 y^{2} = 1

(4)

(2x−y)(6x−3y)−3=0(2 x-y)(6 x-3 y)-3 = 0.

Problem 558

Using Lagrange's method, determine the affine type of the following second-order curves:

?
(1)

2x2+3xy+4y2−5x+2y−1=02 x^{2}+3 x y+4 y^{2}-5 x+2 y-1 = 0;

(2)

4x2−4xy+y2−8x+6y−2=04 x^{2}-4 x y+y^{2}-8 x+6 y-2 = 0;

(3)

2xy−4y2+6x+6y+1=02 x y-4 y^{2}+6 x+6 y+1 = 0;

(4)

x2−4xy+y2−4x+2y−2=0x^{2}-4 x y+y^{2}-4 x+2 y-2 = 0

(5)

x2−2xy+4y2+2x−2y−4=0x^{2}-2 x y+4 y^{2}+2 x-2 y-4 = 0;

(6)

x2+4xy+4y2−6x−8y=0x^{2}+4 x y+4 y^{2}-6 x-8 y = 0;

(7)

2x2−5xy−12y2−x+26y−10=02 x^{2}-5 x y-12 y^{2}-x+26 y-10 = 0;

(8)

3x2+xy−2y2−5x+5y−2=03 x^{2}+x y-2 y^{2}-5 x+5 y-2 = 0;

(9)

4x2+16xy+15y2−8x−22y−5=04 x^{2}+16 x y+15 y^{2}-8 x-22 y-5 = 0;

(10)

4x2−4xy+y2−6x+3y−4=04 x^{2}-4 x y+y^{2}-6 x+3 y-4 = 0.