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Math Help - ellipses

  1. #1
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    ellipses

    Sketch the graphs of the following function:

    9x^2-7x+y^2-4y+112=0


    Any help/or complete working out would be appreciated!
    I have tried to do this and failed.
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  2. #2
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    Use completing the square to change the function into something like:
    9(x - a)^2 + (y - b)^2 = c

    Can you first figure out where the center of the ellipse is?
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  3. #3
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    Hello, Joker37!

    I bet that the "112" should be negative . . . even then, the answer is ugly!

    You were assigned this problem with absolutely no instructions?
    How cruel!


    \text{Sketch the graph of the following function: }\;9x^2-7x+y^2-4y-112\:=\:0

    We have: . 9(x^2 - \frac{7}{9}x \quad) + (y^2 - 4y \quad) \;=\;112

    Complete the square: . 9\left(x^2 - \frac{7}{9}x + \frac{49}{324} - \frac{49}{384}\right) + (y^2 - 4y + 4 - 4) \;=\;112

    . . 9\left(x^2 - \frac{7}{9}x + \frac{49}{324}\right) - \frac{49}{36} + (y^2 - 4y + 4) - 4 \:=\:112

    . . . . . . 9\left(x - \frac{7}{18}\right)^2 + (y - 2)^2 \;=\;112 + \frac{49}{36} + 4

    . . . . . . 9\left(x-\frac{7}{18}\right)^2 + (y-2)^2 \;=\;\frac{4225}{36}


    Divide by \frac{4225}{36}

    . . \displaystyle \frac{9(x-\frac{7}{18})^2}{\frac{4225}{36}} + \frac{(y-2)^2}{\frac{4225}{36}} \;=\;1

    . . \displaystyle \frac{(x - \frac{7}{18})^2}{\frac{4225}{324}} + \frac{(y-2)^2}{\frac{4225}{36}} \;=\;1

    . . \displaystyle \frac{(x-\frac{7}{18})^2}{\left(\frac{65}{18}\right)^2} + \frac{(y-2)^2}{\left(\frac{65}{6}\right)^2} \;=\;1


    \text{Center: }\:\left(\frac{7}{18},\:2\right)

    . . a = \frac{65}{6} \;\text{ (vertical)}

    . . b = \frac{65}{18}\;\text{ (horizontal)}

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  4. #4
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    Actually, as it is written, it is very easy- the graph is the empty set!
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  5. #5
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    Quote Originally Posted by Soroban View Post
    Hello, Joker37!

    I bet that the "112" should be negative . . . even then, the answer is ugly!

    You were assigned this problem with absolutely no instructions?
    How cruel!


    We have: . 9(x^2 - \frac{7}{9}x \quad) + (y^2 - 4y \quad) \;=\;112

    Complete the square: . 9\left(x^2 - \frac{7}{9}x + \frac{49}{324} - \frac{49}{384}\right) + (y^2 - 4y + 4 - 4) \;=\;112

    . . 9\left(x^2 - \frac{7}{9}x + \frac{49}{324}\right) - \frac{49}{36} + (y^2 - 4y + 4) - 4 \:=\:112

    . . . . . . 9\left(x - \frac{7}{18}\right)^2 + (y - 2)^2 \;=\;112 + \frac{49}{36} + 4

    . . . . . . 9\left(x-\frac{7}{18}\right)^2 + (y-2)^2 \;=\;\frac{4225}{36}


    Divide by \frac{4225}{36}

    . . \displaystyle \frac{9(x-\frac{7}{18})^2}{\frac{4225}{36}} + \frac{(y-2)^2}{\frac{4225}{36}} \;=\;1

    . . \displaystyle \frac{(x - \frac{7}{18})^2}{\frac{4225}{324}} + \frac{(y-2)^2}{\frac{4225}{36}} \;=\;1

    . . \displaystyle \frac{(x-\frac{7}{18})^2}{\left(\frac{65}{18}\right)^2} + \frac{(y-2)^2}{\left(\frac{65}{6}\right)^2} \;=\;1


    \text{Center: }\:\left(\frac{7}{18},\:2\right)

    . . a = \frac{65}{6} \;\text{ (vertical)}

    . . b = \frac{65}{18}\;\text{ (horizontal)}
    Haha, well it says +112.
    However the simplified answer for the equation is supposed to be 36(x-4)^2 + 4(y-2)^2 = 144.

    Soroban, I really appreciate your help and efforts so far!
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  6. #6
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    Hi everybody!
    I'm still stuck with this problem and unfortunately I have no-body to help me. I feel so alone.

    Anyway, if it's because the question doesn't make sense or something like that please tell me and I will skip this question but I just don't like knowing that I don't know how to do something.
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  7. #7
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    Quote Originally Posted by Joker37 View Post
    Haha, well it says +112.
    However the simplified answer for the equation is supposed to be 36(x-4)^2 + 4(y-2)^2 = 144.

    Soroban, I really appreciate your help and efforts so far!

    If you look carefully at the equations used, you will notice that 36(x-4)^2 + 4(y-2)^2 = 144 and 9x^2-7x+y^2-4y+112=0 are different equations.


    If you expand and simplify the simplified one, you will get: 9 x^2-72 x+y^2-4 y+112 = 0 which I'm guessing is the equation you are trying to graph and that the one in the first post has a typing error?

    Now that Soroban has supplied you with a method, why don't you give it a try yourself?
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  8. #8
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    Hello, Joker37!

    \text{Haha, well it says +112.}

    \text{However the simplified answer for the equation is supposed to be:}
    . . 36(x-4)^2 + 4(y-2)^2 \:=\: 144

    Well, that certainly explains a lot!


    First of all, the original equation had a typo
    . . It was not 7x, but 72x.


    Second, their answer is quite stupid.
    . . That is definitely not the simplfied answer.

    \begin{array}{ccccccc}\text{Neither is:} &18(x-4)^2 + 2(y-2)^2 \:=\:72 \\<br />
\text{nor:} & 360(x-4)^2 + 40(y-2)^2 \:=\:1440 \end{array}

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