well 發問時間: 科學數學 · 4 年前

Let L be any tangent line to curve x^(0.5)+y^(0.5)=c^(0.5) (in the first quadrant). show that the sum of the x- and y- intercepts of L is c.?

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  • 4 年前
    最佳解答

    Sol

    x^(1/2)+y^(1/2)=c^(1/2)

    (1/2)x^(-1/2)+(1/2)y^(-1/2)y’=0

    x^(-1/2)+y^(-1/2)y’=0

    y^(-1/2)y’=-x^(-1/2)

    y’=-(x/y)^(-1/2)=-(y/x)^(1/2)

    Set L:y-b=-(b/a)^(1/2)*(x-a)

    a^(1/2)+b^(1/2)=c^(1/2)

    When x=0

    y-b=-(b/a)^(1/2)(-a)

    y=b+(ab)^(1/2)

    When y=0

    0-b=-(b/a)^(1/2)*(x-a)

    X-a=(ab)^(1/2)

    x=a+(ab)^(1/2)

    So

    [b+(ab)^(1/2)]+[a+(ab)^(1/2)]

    =a+b+2(ab)^(1/2)

    =[a^(1/2)+b^(1/2)]^2

    =[c^(1/2)]^2

    =c

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