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    i am stuck on this question and was wondering if i could get any help:

    the equation of a circle is x^2 + y^2 = 1. a curve is defined as C' = f(C), which is the image of C under the linear transformation f represented by the matrix:

    2 1
    1 2

    i.e. the inverse image of C' (the curve) is C (the circle).

    Find the equation of the curve C'.

    i have a linear transformation of f(x,y) = (2x + y, x + 2y) and i can also work out the inverse and determinant of the matrix. but i dont know how to put it all together to get the equation of the curve help is much appreciated
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    Hmm working from first principles here ..

    If you had a point (x,y) on C', you could work out from which point on C it had appeared, by inverting your matrix.

    If that point were (p,q), you would know that p^2 + q^2 = 1 for it to be on C.

    Is that enough??
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    if you're allowed to use a parametrisation you can sub x for cost and y for sint and set 0<t<2pi
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    (Original post by ian.slater)
    Hmm working from first principles here ..

    If you had a point (x,y) on C', you could work out from which point on C it had appeared, by inverting your matrix.

    If that point were (p,q), you would know that p^2 + q^2 = 1 for it to be on C.

    Is that enough??
    i am not following:o: i cant do that because i dont have the equation of the curve. can you give a bit more help?
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    By inverting your matrix, you can find out the point that was transformed to say (x,y)

    (p,q) = M^-1 (x,y)

    So you have p and q in terms of x and y.

    For (p,q) to be on C, p^2 + q^2 = 1. Which is now an equation connecting x and y.

    That equation is your C'
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    (Original post by ian.slater)
    By inverting your matrix, you can find out the point that was transformed to say (x,y)

    (p,q) = M^-1 (x,y)

    So you have p and q in terms of x and y.

    For (p,q) to be on C, p^2 + q^2 = 1. Which is now an equation connecting x and y.

    That equation is your C'
    thanks for the help, i eventually got it. can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0? thanks again:o:
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    (Original post by wrooru)
    can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0?
    Agreed.
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    (Original post by wrooru)
    thanks for the help, i eventually got it. can you quickly check for me to see if the answer is 5x^2 + 5y^2 - 8xy - 9 = 0? thanks again:o:
    I get the same answer. I'm also wondering whether the same result can be obtained by thinking geometrically - i.e. what does the (2,1/1,2) matrix represent in terms of shear and stretch?
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    I get eigenvalues of 1 and 3. The eigenvector corresponding to 3 is (1,-1). So the transformation is a stretch of factor 3 along the axis y=-x

    That leads to 9(x+y)^2 + (x-y)^2 = 18 which is the same equation. Seems like more work though.
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