Question

Show that the four asymptotes of the curve (x ^ 2 - y ^ 2)(y ^ 2 - 4x ^ 2) + 6x ^ 3 - 5x ^ 2 * y - 3x * y ^ 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 .

Answer 1

Answer:

We have x + 2y ≤ 3, 3x + 4y > 12, x > 0, y ≥ 1

Now let’s plot lines x + 2y = 3, 3x + 4y = 12, x = 0 and y = 1 in coordinate plane.

Line x + 2y = 3 passes through the points (0, 3/2) and (3, 0).

Line 3jc + 4y = 12 passes through points (4, 0) and (0, 3).

For (0, 0), 0 + 2(0) – 3 < 0.

Therefore, the region satisfying the inequality x + 2y ≤ 3 and (0,0) lie on the same side of the line x + 2y = 3.

For (0, 0), 3(0) + 4(0)- 12 ≤0.

Therefore, the region satisfying the inequality 3x + 4y ≥ 12 and (0, 0) lie on the opposite side of the line 3x + 4y = 12.

The region satisfying x > 0 lies to the right hand side of the y-axis.