彭羿凡 發問時間: 科學數學 · 4 年前

請教 微積分 let L be any tangent line to the curve X^1/2+Y^1/2=C^1/2 (in the first quadrant). The sum of the x- and y-intercepts of L is ?

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    Question:

    Let L be any tangent line to the curve √x + √y = √c (in the first quadrant).

    The sum of the x- and y-intercepts of L is ?

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    Solution:

    √x + √y = √c

    Differentiate both sides with respect to x ...... [ Implicit Differentiation ]

    1/(2√x) + 1/(2√y) dy/dx = 0

    dy/dx = -√(y/x)

    The equation of the tangent line L at (x₀,y₀):

    y - y₀ = -√(y₀/x₀) (x - x₀)

    y = -√(y₀/x₀) (x - x₀) + y₀

    The x-intercept of L = x₀ + √(x₀ y₀)

    The y-intercept of L = y₀ + √(x₀ y₀)

    [ √x + √y = √c    ]

    [ x + y + 2√(xy) = c ]

    The sum of the x- and y-intercepts of L is x₀ + y₀ + 2√(x₀ y₀) = c

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