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Show that the four asymptotes of the curve (x2 . 10. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is . . -y2)(y2-4x2) 6x3 – 5x2y – 3xy2 2y3 – x2 3xy – 1 = 0 Cut the curve again in eight points which lie on the circle x2 y2 = 1?
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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.

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Show that the four asymptotes of the curve (x2 . 10. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is . . -y2)(y2-4x2) 6x3 – 5x2y – 3xy2 2y3 – x2 3xy – 1 = 0 Cut the curve again in eight points which lie on the circle x2 y2 = 1?
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Show that the four asymptotes of the curve (x2 . 10. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is . . -y2)(y2-4x2) 6x3 – 5x2y – 3xy2 2y3 – x2 3xy – 1 = 0 Cut the curve again in eight points which lie on the circle x2 y2 = 1? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Show that the four asymptotes of the curve (x2 . 10. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is . . -y2)(y2-4x2) 6x3 – 5x2y – 3xy2 2y3 – x2 3xy – 1 = 0 Cut the curve again in eight points which lie on the circle x2 y2 = 1? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that the four asymptotes of the curve (x2 . 10. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is . . -y2)(y2-4x2) 6x3 – 5x2y – 3xy2 2y3 – x2 3xy – 1 = 0 Cut the curve again in eight points which lie on the circle x2 y2 = 1?.
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