Anonymous
Anonymous asked in Science & MathematicsMathematics · 6 months ago

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

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  • 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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