How do I find the exact coordinates of all points xy where the line x y intersects the curve x^2 y^2 - 2y^2 4x^2 - y^2?
In this article, we will explore the process of finding the intersection points between a line and a curve. Specifically, we will be dealing with the line yequalsx and the curve x^2 y^2 - 2y^2 4x^2 - y^2.
By substituting y x into the given curve equation, we aim to solve for the values of x and subsequently determine the co-ordinates of the intersection points.
Substitution and Simplification
The given equation is: x^2 y^2 - 2y^2 4x^2 - y^2
Substituting y x into the curve equation:
x^2 x^2 - 2x^2 4x^2 - x^2
This simplifies to:
yields 2x^2 - 2x^2 42x^2
Continuing, we simplify further to:
x - 1^2 2
Using the Quadratic Formula
Taking the square root of both sides, we obtain:
x - 1 sqrt{2}, quad x - 1 -sqrt{2}
This results in:
x sqrt{2} 1, quad x -sqrt{2} - 1
Finding Corresponding y Values
Since y x, we find the corresponding y values as follows:
- For x sqrt{2} 1, y sqrt{2} 1
- For x -sqrt{2} - 1, y -sqrt{2} - 1
Special Case: x 0
When x 0, y 0 as well. Substituting 0 0 into the curve equation:
0^2 0^2 - 2 cdot 0^2 40^2 - 0^2
This simplifies to:
0 0
Hence, the point (0, 0) is also a valid intersection point.
Conclusion
The intersection points determined are:
(sqrt{2} 1, sqrt{2} 1) (-sqrt{2} - 1, -sqrt{2} - 1) (0, 0)These points represent the exact coordinates of the intersection between the given line and curve. To summarize, the process involves substituting the line equation into the curve and using algebraic techniques to solve for the intersection points, such as factoring and the quadratic formula.
General Approach
For similar problems where a line intersects a curve, follow these general steps:
Substitute the line equation (e.g., y x) into the curve equation. Simplify the resulting equation. Solve for one variable (e.g., x). Substitute the solutions back into the line equation to find the corresponding y values. Handle any special cases, such as when the line is vertical (e.g., x 0).By employing these steps, you can systematically find the intersection points and ensure accurate solutions.
Keywords
intersection points, line equation, curve equation, algebraic substitution