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Math Help - Tangent of ellipse

  1. #1
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    Tangent of ellipse

    The tangent at P(4\cos\theta, 3\sin\theta) on the ellipse 9x^2+16y^2=144 meets the tangent at the positive end of the major axis at Q and the positive end of the minor axis at R. Find
    a) the ratio of the lengths PQ and PR
    b) the parametric equations of the midpoint of QR.

    The equation of the tangent at the point P is bx\cos\theta+ay\sin\theta=ab
    plugging the values i got
    3x\cos\theta+4y\sin\theta=12
    then i found the point where it met the axes
    P(4\sec\theta,0)
    and Q(0,3\csc\theta)
    The i found PQ and PR
    PQ=\sqrt{(4\cos\theta-4\sec\theta)^2+(3\sin\theta)^2}
    =\sin\theta\sqrt{16\tan^2\theta+9}
    and PR=\sqrt{(4\cos\theta)^2+(3\sin\theta-3\csc\theta)^2}
    =\frac{\cos\theta}{\tan\theta}\sqrt{16\tan^2\theta  +9}
    then the ratio would be \sin\theta:\frac{\cos\theta}{\tan\theta}
    but the answer is \cos\theta(1-\cos\theta):\sin\theta(1-\sin\theta)
    Thanks for any help!
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  2. #2
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    Quote Originally Posted by arze View Post
    The tangent at P(4\cos\theta, 3\sin\theta) on the ellipse 9x^2+16y^2=144 meets the tangent at the positive end of the major axis at Q and the positive end of the minor axis at R. Find
    a) the ratio of the lengths PQ and PR
    b) the parametric equations of the midpoint of QR.

    The equation of the tangent at the point P is bx\cos\theta+ay\sin\theta=ab
    plugging the values i got
    3x\cos\theta+4y\sin\theta=12
    then i found the point where it met the axes
    P(4\sec\theta,0)
    and Q(0,3\csc\theta)
    The i found PQ and PR
    PQ=\sqrt{(4\cos\theta-4\sec\theta)^2+(3\sin\theta)^2}
    =\sin\theta\sqrt{16\tan^2\theta+9}
    and PR=\sqrt{(4\cos\theta)^2+(3\sin\theta-3\csc\theta)^2}
    =\frac{\cos\theta}{\tan\theta}\sqrt{16\tan^2\theta  +9}
    then the ratio would be \sin\theta:\frac{\cos\theta}{\tan\theta}
    but the answer is \cos\theta(1-\cos\theta):\sin\theta(1-\sin\theta)
    You have taken P and Q to be the points where the tangent meets the coordinate axes. But the question says that P and Q are the points where the tangent meets the axes of the ellipse, namely the lines x=4 and y=3.
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  3. #3
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    sorry, i thought the major and minor axes where the center was at the origin would be the x and y axes?
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  4. #4
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    Quote Originally Posted by arze View Post
    sorry, i thought the major and minor axes where the center was at the origin would be the x and y axes?
    My mistake, I should have said what the question says, which is:
    The tangent at P(4\cos\theta, 3\sin\theta) on the ellipse 9x^2+16y^2=144 meets the tangent at the positive end of the major axis at Q and the positive end of the minor axis at R.
    The last part of that sentence is a bit misleading. It must mean that the tangent at P meets the tangent at the positive end of the minor axis at R. You need to take Q and R to be the points where the tangent at P meets the lines x=4 and y=3.
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  5. #5
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    Ok i understand know, haha my mistake. Thanks!
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