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# What are the 2 x-coordinates?

Find all the x-coordinates (in increasing order) of the points on the curve x^2y^2+xy=2 where the slope of the tangent line is −1.

### 1 Answer

- 6 months ago
x^2 * y^2 + xy = 2

(xy)^2 + (xy) = 2

2 * (xy) * (x * dy + y * dx) + (x * dy + y * dx) = 0

(2 * xy + 1) * (x * dy + y * dx) = 0

(2xy + 1) * (x * dy/dx + y) = 0

dy/dx = -1

(2xy + 1) * (x * -1 + y) = 0

(2xy + 1) * (y - x) = 0

2xy + 1 = 0

2xy = -1

xy = -1/2

x = -1/(2y)

y - x = 0

y = x

(xy)^2 + xy = 2

x = -1/(2y)

(-1 * y / (2y))^2 + (-1/(2y)) * y = 2

(-1/2)^2 + (-1/2) = 2

1/4 - 1/2 = 2

Doesn't fit, so that's extraneous

(xy)^2 + (xy) = 2

y = x

(x * x)^2 + (x * x) = 2

x^4 + x^2 = 2

x^4 + x^2 - 2 = 0

x^2 = (-1 +/- sqrt(1 + 8)) / 2

x^2 = (-1 +/- 3) / 2

x^2 = -4/2 , 2/2

x^2 = -2 , 1

x^2 = -2 is not defined in the real number system

x^2 = 1

x = -1 , 1

(-1 , -1) and (1 , 1)

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