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Albert1
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$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$
prove $(1)$ has no solution
prove $(1)$ has no solution
Albert said:$x,y\in N,\,\, and \,\,\dfrac {1}{x^2}+\dfrac{1}{xy}+\dfrac {1}{y^2}=1----(1)$
prove $(1)$ has no solution
The equation "1/x^2+1/xy+1/y^2=1" is a mathematical expression that represents a relationship between three variables, x, y, and a constant value of 1. It is known as a quadratic equation in two variables with a constant term.
To prove that the equation "1/x^2+1/xy+1/y^2=1" has no real solutions, we can use the discriminant, which is a mathematical tool used to determine the nature of the roots of a quadratic equation. In this case, the discriminant will be less than zero, indicating that the equation has no real solutions.
Yes, for example, if we substitute x=1 and y=2 into the equation "1/x^2+1/xy+1/y^2=1", we get 1/1+1/2+1/4=1, which simplifies to 3/4=1, which is not a true statement. This shows that the equation does not have any real solutions.
Yes, the equation "1/x^2+1/xy+1/y^2=1" represents a conic section known as a hyperbola. It can be graphed on a coordinate plane and will have a vertical and horizontal asymptote, indicating that it does not intersect the x and y axes, further confirming the lack of real solutions.
Yes, the equation "1/x^2+1/xy+1/y^2=1" can have complex solutions. In fact, it will have two complex solutions for every set of x and y values that satisfy the equation. This is due to the nature of quadratic equations and their solutions in the complex number system.