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Math Help - differentiation - tangent meeting another pt on y=x^3

  1. #1
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    differentiation - tangent meeting another pt on y=x^3

    bk2 p41 q48
    question : given the curve C: y = x^3 and P(h,k) is a point on C , where h and k are non-zero no.
    (a)find the equation of the tangent to C at P
    (b) if the tangent found in (a) intersects C again at Q , find the coordinates of Q.

    my working:
    dy/dx = 3x^2
    equ. : y= 3h^2 (x-h)+k
    sub to C
    x^3 - 3h ^2 x +3h^3 -k = 0
    don't know how to solve and find Q
    thanks!
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Never mind I cant take a derivative
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  3. #3
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    Isomorphism's Avatar
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    Quote Originally Posted by afeasfaerw23231233 View Post
    bk2 p41 q48
    question : given the curve C: y = x^3 and P(h,k) is a point on C , where h and k are non-zero no.
    (a)find the equation of the tangent to C at P
    (b) if the tangent found in (a) intersects C again at Q , find the coordinates of Q.

    my working:
    dy/dx = 3x^2
    equ. : y= 3h^2 (x-h)+k
    sub to C
    x^3 - 3h ^2 x +3h^3 -k = 0
    don't know how to solve and find Q
    thanks!
    Since P lies on C, k = h^3

    x^3 - 3h ^2 x +3h^3 - k = 0 \Rightarrow x^3 - 3h ^2 x +2h^3= 0

    Now since P satisfies the line as well as the curve already, h is a root of the cubic. By this and by long division:

    x^3 - 3h ^2 x +2h^3= 0 \Rightarrow (x - h)(x^2 + hx - 2h^2)= 0 \Rightarrow (x - h)^2(x+2h) = 0

    So finish it
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