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Math Help - simultaneous equations

  1. #1
    Junior Member
    Joined
    Oct 2007
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    simultaneous equations

    completely stuck here:

    1. find the value of k for which y = 3x+1 is a tangent to the curve
    x^2 + y^2 = k

    2. find the range of values of k for which y=x-3 meets x^2-3y^2=k in two distinct points.

    and i went wrong in this question, no idea where though

    find the number of points of intersection
    y= x/4 + 1
    y^2 = x

    i multiplied by 4 to get rid of the fraction. do i square root x ??
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  2. #2
    Super Member

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    Hello, lra11

    2. Find the range of values of k for which y\:=\:x-3
    meets x^2-3y^2\:=\:k in two distinct points.
    Find their intersections; substitute the first equation into the second.

    . . x^2 - 3(x - 3)^2 \:=\:k\quad\Rightarrow\quad 2x^2 - 8x + (k+27) \:=\:0


    Quadratic Formula: . x \;=\;\frac{-(-8)\pm\sqrt{(-8)^2 - (4)(2)(k+27)}}{2(2)} \;=\;\frac{8 \pm\sqrt{-8k-152}}{4}

    A quadratic equation has two distinct roots if its Discriminant is positive.

    So we have: . -8k - 152 \:> \:0\quad\Rightarrow\quad -8k \:> \:152\quad\Rightarrow\quad k \:< \:-19


    Therefore, for two distinct intersections: .  k \in (-\infty,\;-19)

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