Question

X^2-y^2=18 { xy=9.

Answer 1

x=9/y

81-y^4=18y^2

y^4+18y^2+81=162

(y^2+9)^2=9^2*2

y^2=9*(sqrt(2)-1)

x^2=9*(sqrt(2)+1)

Решения:

х=3*sqrt(sqrt(2)+1))

y=3*sqrt(sqrt(2)-1))

или

х=-3*sqrt(sqrt(2)-1))

y=-3*sqrt(sqrt(2)+1))

Answer 2

To solve this system of equations, we can use the method of substitution.

First, let's solve for one variable in terms of the other from the second equation. We can rearrange xy = 9 to get y = 9/x.

Substitute this expression for y into the first equation:

x^2 - (9/x)^2 = 18

Simplify and multiply both sides by x^2 to get rid of the fraction:

x^4 - 18x^2 - 81 = 0

This is a quadratic equation in x^2. We can solve for x^2 using the quadratic formula:

x^2 = (18 ± sqrt(18^2 + 4*81))/2

x^2 = (18 ± sqrt(684))/2

x^2 = 9 ± 3sqrt(19)

Therefore, we have two solutions for x: x = sqrt(9 + 3sqrt(19)) and x = -sqrt(9 - 3sqrt(19)).

Substitute each value of x into the equation y = 9/x to find the corresponding values of y:

If x = sqrt(9 + 3sqrt(19)), then y = 9/x = 9/sqrt(9 + 3sqrt(19)).

If x = -sqrt(9 - 3sqrt(19)), then y = 9/x = -9/sqrt(9 - 3sqrt(19)).

So the two solutions to the system of equations are:

(sqrt(9 + 3sqrt(19)), 9/sqrt(9 + 3sqrt(19))) and (-sqrt(9 - 3sqrt(19)), -9/sqrt(9 - 3sqrt(19))).