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Math Help - Coordinate Geometry Ellipse

  1. #1
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    Unhappy Coordinate Geometry Ellipse

    Please help me solve this

    Q: is the midpoint from directrix to C the focus?


    determine the equation and graph of the ellipse w/ center at (4,-5), major axis vertical. eccentricity=√21/5 distance between directrices=50√21/21


    (note: distance from C to a directrix = a/e or a2/c) note (e = c / a)


    Thanks in advance for the reply ^^
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  2. #2
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    Re: Coordinate Geometry Ellipse

    i think its not the midpoint
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  3. #3
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    Re: Coordinate Geometry Ellipse

    Do you mean parabola?
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  4. #4
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by mventurina1 View Post
    Please help me solve this

    Q: is the midpoint from directrix to C the focus?


    determine the equation and graph of the ellipse w/ center at (4,-5), major axis vertical. eccentricity=√21/5 distance between directrices=50√21/21

    (note: distance from C to a directrix = a/e or a2/c) note (e = c / a)

    Thanks in advance for the reply ^^
    Good morning,

    1. Let d denotes the distance from C to the directrix. Then you know d = \frac{25}{21} \cdot \sqrt{21}

    With a = e \cdot d follows: a = \frac{\sqrt{21}}{5} \cdot \frac{25}{21} \cdot \sqrt{21} = 5

    2. Since \frac{a^2-b^2}{a^2} = \frac{21}{25}~\implies~b = 2

    3. Therefore the equation of the ellipse is:

    \frac{(x-4)^2}4+\frac{(y+5)^2}{25}=1

    4. The coordinates of the foci are F_1\left(4\ ,\ -5+\sqrt{21} \right) and F_2\left(4\ ,\ -5-\sqrt{21} \right)

    5. The equation of the upper directrix is d:y = -5+\frac{25}{21} \cdot \sqrt{21}

    6. The major axis is placed on the straight line a (see attachment). d crosses a in the point D\left(4, -5+\frac{25}{21} \cdot \sqrt{21} \right)

    7. The midpoint between C and D is M \left(4, -5+\frac{25}{42} \cdot \sqrt{21} \right) which is definitely not the focus F (compare the result at #4)
    Attached Thumbnails Attached Thumbnails Coordinate Geometry Ellipse-ellipmitdirectrix.png  
    Thanks from mventurina1
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  5. #5
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    Re: Coordinate Geometry Ellipse

    ellipse
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by earboth View Post
    Good morning,

    1. Let d denotes the distance from C to the directrix. Then you know d = \frac{25}{21} \cdot \sqrt{21}

    With a = e \cdot d follows: a = \frac{\sqrt{21}}{5} \cdot \frac{25}{21} \cdot \sqrt{21} = 5

    2. Since \frac{a^2-b^2}{a^2} = \frac{21}{25}~\implies~b = 2

    3. Therefore the equation of the ellipse is:

    \frac{(x-4)^2}4+\frac{(y+5)^2}{25}=1

    4. The coordinates of the foci are F_1\left(4\ ,\ -5+\sqrt{21} \right) and F_2\left(4\ ,\ -5-\sqrt{21} \right)

    5. The equation of the upper directrix is d:y = -5+\frac{25}{21} \cdot \sqrt{21}

    6. The major axis is placed on the straight line a (see attachment). d crosses a in the point D\left(4, -5+\frac{25}{21} \cdot \sqrt{21} \right)

    7. The midpoint between C and D is M \left(4, -5+\frac{25}{42} \cdot \sqrt{21} \right) which is definitely not the focus F (compare the result at #4)



    thank you sir for the answer !!! now I understand
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  7. #7
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by mventurina1 View Post
    thank you sir for the answer !!! now I understand
    Hello,

    I noticed to late that there is a second solution.

    There exists a second directrix whose equation is
    d:y = -5-\frac{25}{21} \cdot \sqrt{21}

    This directrix crosses the straight line a in P\left(4, -5-\frac{25}{21} \cdot \sqrt{21} \right)

    The midpoint between F and P is indeed C. So the statement in the question is true under certain conditions.
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  8. #8
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by earboth View Post
    Hello,

    I noticed to late that there is a second solution.

    There exists a second directrix whose equation is
    d:y = -5-\frac{25}{21} \cdot \sqrt{21}

    This directrix crosses the straight line a in P\left(4, -5-\frac{25}{21} \cdot \sqrt{21} \right)

    The midpoint between F and P is indeed C. So the statement in the question is true under certain conditions.

    you mean under certain conditions . the focus is the midpoint of the directrix and center?
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  9. #9
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by mventurina1 View Post
    you mean under certain conditions . the focus is the midpoint of the directrix and center?
    Hello,

    please do me a favor and forget my last answer - it is wrong. I must have had a brain blackout. Sorry for the confusion!
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  10. #10
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    Re: Coordinate Geometry Ellipse

    Quote Originally Posted by earboth View Post
    Hello,

    please do me a favor and forget my last answer - it is wrong. I must have had a brain blackout. Sorry for the confusion!
    Sure and thanks again Sir !
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