Question

Cистема: x^2-y^2=2a+3 xy=a+1При каких a система имеет два решения?

Answer 1

при а больше , либо равно нулю.

Answer 2

Solution for the System x^2 - y^2 = 2a + 3 and xy = a + 1

To find the values of a for which the system has two solutions, we can use the discriminant of the quadratic equation formed by the given system.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. If the discriminant is greater than zero, the quadratic equation has two distinct real solutions.

Calculating the Discriminant for the Given System

The given system of equations is: 1. x^2 - y^2 = 2a + 3 2. xy = a + 1

From equation (2), we can express x in terms of y as: x = (a + 1) / y.

Substituting this value of x into equation (1), we get: ((a + 1) / y)^2 - y^2 = 2a + 3

Multiplying through by y^2 to clear the fraction, we get: (a + 1)^2 - y^4 = (2a + 3)y^2

This equation is a quadratic in y^2. The discriminant of this quadratic equation will help us determine the values of a for which the system has two solutions.

Finding the Values of a for Two Solutions

The discriminant of the quadratic equation (a + 1)^2 - y^4 = (2a + 3)y^2 is given by D = (2a + 3)^2 - 4((a + 1)^2)(-1).

For the system to have two solutions, the discriminant D must be greater than zero.

Conclusion

By solving the inequality (2a + 3)^2 - 4((a + 1)^2)(-1) > 0, we can find the values of a for which the given system has two solutions.