Find all points on the curve x^2y^2+2xy=3 where the slope of the tangent is -1. If there is more than one point, use a comma to separate each pair of coordinates
Do you mean the curve x^2 + y^2 + 2xy = 3?
On second thoughts, I don't think you do mean that: if you did, the tangent would be -1 for every point on the curve.
If it IS x²y² + 2xy = 3, then put z=xy for a moment, in which case z²+2z=3, in which case z=1 or z=-3. So xy=1 or xy=-3, i.e. y=1/x or y=-3/x. If you differentiate each with respect to x and solve for dy/dx = -1, you should get the answers you need.
Big juicy pickle
To find the points on the curve where the slope of the tangent is -1, we need to use calculus. First, let's rewrite the equation of the curve in terms of y:
x^2y^2 + 2xy = 3
y^2(x^2 + 2x) = 3
y^2 = 3 / (x^2 + 2x)
Now, to find the slope of the tangent, we need to take the derivative of y with respect to x:
dy/dx = [d/dx (3 / (x^2 + 2x))]^(1/2)
= [d/dx (3(x^2 + 2x))^(-1)]^(1/2)
= [3(-2x - 2) / (x^2 + 2x)^2]^(1/2)
= [(6 - 6x) / (x^2 + 2x)^2]^(1/2)
= (6 - 6x)^(1/2) / (x^2 + 2x)
Now, we need to set the derivative equal to -1 and solve for x:
(6 - 6x)^(1/2) / (x^2 + 2x) = -1
Let's square both sides to eliminate the square root:
[(6 - 6x)^(1/2) / (x^2 + 2x)]^2 = (-1)^2
(6 - 6x) / (x^2 + 2x)^2 = 1
Now, cross multiply and rearrange the equation to a quadratic form:
6 - 6x = x^2 + 2x^3 + 4x^2
0 = 2x^3 + 4x^2 + 6x - 6x - x^2 - 6
Simplifying further:
2x^3 + 3x^2 - 6 = 0
At this point, it is not easy to solve this cubic equation analytically. Therefore, we can use numerical methods or software to find the approximate values of x.
Once we have the values of x, we can substitute them back into the equation y^2 = 3 / (x^2 + 2x) to find the corresponding values of y.
Finally, list all the pairs of coordinates (x, y) for which the slope of the tangent is -1.