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Math Help - Need help with these 2 conic sections problems

  1. #1
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    Need help with these 2 conic sections problems

    Idk if this belongs in the pre calc forum, but im in pre calc and ym teacher assigned conic sections as an independent chapter. If this needs to be moved, please move it to the correct forum.

    #1) Find the equation of a hyperbola with foci (-4,2) and (2,2) and slope of one asymptote is 3/5




    #2) solve the equation for y and graph: 2x^2-xy+3y^2-3x+4y-6=0

    Any help would be great. Thanks in advance to whoever tries to help!
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  2. #2
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    Hello, Zdmey!

    There is a typo is #1 . . . the numbers don't fit.


    1) Find the equation of a hyperbola with foci (-4,2) and (2,2)
    and slope of one asymptote is 3/5 . . Could this be 4/5 ?

    The foci are horizontally oriented.
    Hence, the hyperbola is "horizontal": . \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} \:=\:1

    The center is halfway between the foci: . \boxed{(h,k) \:=\:(-1,2)}
    . . And: . c = 3


    The slope of the asymptotes is: . m \:=\:\pm\frac{b}{a}

    If this slope is {\color{red}\frac{4}{5}}, then we have: . \frac{b}{a} \:=\:\frac{4}{5}

    Then a is a multiple of 5, b is a multiple of 4: . a = 5k,\;b = 4k


    Sincce a^2 \:=\:b^2+c^2, we have: . (5k)^2 \:=\:(4k)^2 + 3^2 \quad\Rightarrow\quad 9k^2 \:=\:9 \quad\Rightarrow\quad k \:=\:\pm 1
    . . Hence: . \boxed{a = 5,\;b = 4}


    The equation is: . \frac{(x+1)^2}{25} - \frac{(y-2)^2}{16} \;=\;1

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