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Math Help - Problem in Polar coordinates

  1. #1
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    Problem in Polar coordinates

    1. The curve C has polar equation
    r^2 cos 2Ѳ = a^2, -pi/4 < Ѳ < pi/4
    Find in terms of the positive constant a
    (a) the area of the region bounded by the half-lines Ѳ = + pi/6 and the arc of the curve C in the interval -pi/6 < Ѳ < pi/6
    (b) the equation of the tangent to the curve which is perpendicular to the initial line, giving your answer in polar form.


    2. The curve C has polar equation r = a sin Ѳ sin 2Ѳ, where a is a positive constant and 0 ≤ Ѳ ≤ pi/2.
    (a) Find, in terms of a, the area of the region enclosed by C.
    (b) Show that the tangent at the point (3/4a, pi/3) is parallel to the initial line.


    Please help me to solve these questions. Please help me.
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  2. #2
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    Quote Originally Posted by geton View Post
    1. The curve C has polar equation
    r^2 cos 2Ѳ = a^2, -pi/4 < Ѳ < pi/4
    Find in terms of the positive constant a
    (a) the area of the region bounded by the half-lines Ѳ = + pi/6 and the arc of the curve C in the interval -pi/6 < Ѳ < pi/6
    (b) the equation of the tangent to the curve which is perpendicular to the initial line, giving your answer in polar form.


    2. The curve C has polar equation r = a sin Ѳ sin 2Ѳ, where a is a positive constant and 0 ≤ Ѳ ≤ pi/2.
    (a) Find, in terms of a, the area of the region enclosed by C.
    (b) Show that the tangent at the point (3/4a, pi/3) is parallel to the initial line.


    Please help me to solve these questions. Please help me.
    1. (a) A = \frac{1}{2} \int_{-\pi/6}^{\pi/6} \frac{a^2}{\cos (2\theta)} \, d \theta.

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    2. Area of single 'leaf': A = \frac{1}{2} \int_{0}^{\pi/2} a \sin \theta \sin (2\theta) \, d \theta.

    Area of both 'leaves' (by symmetry): A = \int_{0}^{\pi/2} a \sin \theta \sin (2\theta) \, d \theta and you should substitute \sin \left( 2 \theta \right) = 2 \sin \theta \cos \theta.

    For (b) of both questions, I've got the same questions as in my other reply. What's meant by initial line etc.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    I've got the same questions as in my other reply. What's meant by initial line etc.

    Initial line is l, means x-axis. Thank you for your help. Iíve solved those questions. It is from polar coordinates chapter. And your areaís formula is absolutely right. Lately I understood that chapter. Thank you.
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