Show that the four asymptotes of the curve (x2 . 10. Prove that the le...
Problem:
Show that the four asymptotes of the curve (x^2 - y^2)(y^2 - 4x^2) + 6x^3 - 5x^2y - 3xy^2 + 2y^3 - x^2 + 3xy - 1 = 0 cut the curve again in eight points which lie on the circle x^2 + y^2 = 1.
Solution:
To find the asymptotes of the given curve, we need to factorize the equation and determine the values of x and y that make each factor equal to zero. The equation can be rewritten as follows:
(x^2 - y^2)(y^2 - 4x^2) + 6x^3 - 5x^2y - 3xy^2 + 2y^3 - x^2 + 3xy - 1 = 0
Now, let's factorize the equation:
((x - y)(x + y))(y^2 - 4x^2) + 6x^3 - 5x^2y - 3xy^2 + 2y^3 - x^2 + 3xy - 1 = 0
(x - y)(x + y)(y - 2x)(y + 2x) + 6x^3 - 5x^2y - 3xy^2 + 2y^3 - x^2 + 3xy - 1 = 0
Now, let's consider each factor separately:
1. (x - y) = 0
x = y
2. (x + y) = 0
x = -y
3. (y - 2x) = 0
y = 2x
4. (y + 2x) = 0
y = -2x
Asymptotes:
The asymptotes of the curve are the lines defined by the equations x = y, x = -y, y = 2x, and y = -2x.
Now, we need to find the eight points where the asymptotes cut the curve and check if they lie on the circle x^2 + y^2 = 1.
Point 1: (x, y) = (1, 1)
- x = y
- x^2 + y^2 = 2
- 1 + 1 = 2
- The point lies on the circle x^2 + y^2 = 1.
Point 2: (x, y) = (-1, -1)
- x = y
- x^2 + y^2 = 2
- 1 + 1 = 2
- The point lies on the circle x^2 + y^2 = 1.
Point 3: (x, y) = (1, -1)
- x = -y
- x^2 + y^2 = 2
- 1 + 1 = 2
- The point lies on the circle x^2 + y^2 = 1.