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Math Help - [SOLVED] Length of the major axis?

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    [SOLVED] Length of the major axis?

    heyy, (Barron SAT Math II Page 19)
    what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

    a) 3.16
    b)4.47
    c) 6.32
    d) 8.94
    e) 14.14

    The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 sqrt20. Can please explain why? What does it mean by 'major' axis ?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by fabxx View Post
    heyy, (Barron SAT Math II Page 19)
    what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

    a) 3.16
    b)4.47
    c) 6.32
    d) 8.94
    e) 14.14

    The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 sqrt20. Can please explain why? What does it mean by 'major' axis ?

    Thanks in advance!
    An ellipse with it's center at the origin and the semiaxes a (=major semiaxis) and b(= minor semiaxis) has the equation:

    \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1

    With the given equation you can easily calculate a. This value has to be doubled to get the complete major axis:

    10x^2 + 20y^2 = 200~\implies~\dfrac{x^2}{20} + \dfrac{y^2}{10}=1

    Therefore a^2 = 20~\implies~a=\sqrt{20}=2\sqrt{5}

    The complete axis has the length 2a = 2\cdot 2\sqrt{5} = 4 \cdot \sqrt{5}
    Attached Thumbnails Attached Thumbnails [SOLVED] Length of the major axis?-ellips_axes.png  
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    Grand Panjandrum
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    Quote Originally Posted by fabxx View Post
    heyy, (Barron SAT Math II Page 19)
    what is the length of the major axis of the ellipse whose equation is 10x^2+20y^2=200?

    a) 3.16
    b)4.47
    c) 6.32
    d) 8.94
    e) 14.14

    The correct answer is D). The answer book said to divide both sides of the equation by 200 to write the equation in standard form. So then the length of the major axis is 2 sqrt20. Can please explain why? What does it mean by 'major' axis ?

    Thanks in advance!
    The standard for of the equation of an ellipse is:

    \frac{x^2}{a^2}+\frac{y^2}{b^2}=1

    where the long axis is oriented along the x-axis ond the short along the y-axis.

    The major axis is the long axis, and the minor axis is the short axis. a is half the length of the major axis and b half the length of the minor axis.

    CB
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    Thanks for the answers. But how do you know that a is only half the axis? Thanks in advance
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    Grand Panjandrum
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    Quote Originally Posted by fabxx View Post
    Thanks for the answers. But how do you know that a is only half the axis? Thanks in advance
    Because when y=0:

    x^2=a^2

    so:

    x=\pm a

    are the two points where the ellipse cuts the x axis, the distance between them 2a, is the length of the major axis.

    CB
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