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Math Help - one last question...

  1. #1
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    one last question...

    I have been given two questions, both of which i ahve no clue on what to do

    linear transformation T : R^2-->R^2 with standard matrix M=[a,0;0,b] a>b>0

    a) show the circle x^2+y^2=r^2 is transformed by T to the ellipse:

    (x^2)/(a^2)+(y^2)/(b^2)=r^2

    hint, let P(rcos(theta), rsin(theta)) be a point on the circle and show that P is transformed to Q(arcos(theta), brsin(theta)), a point on the ellipse.

    b)derive a formula for the area of an ellipse
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  2. #2
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    Quote Originally Posted by deragon999 View Post
    I have been given two questions, both of which i ahve no clue on what to do

    linear transformation T : R^2-->R^2 with standard matrix M=[a,0;0,b] a>b>0

    a) show the circle x^2+y^2=r^2 is transformed by T to the ellipse:

    (x^2)/(a^2)+(y^2)/(b^2)=r^2

    hint, let P(rcos(theta), rsin(theta)) be a point on the circle and show that P is transformed to Q(arcos(theta), brsin(theta)), a point on the ellipse.

    b)derive a formula for the area of an ellipse
    a) Find the image of (r \cos \theta, ~ r \sin \theta) in the following way:

    Let X be the column matrix [r \cos \theta; ~ r \sin \theta]. Calculate MX .....

    b) If you operate on an object with a linear transformation, the area of the image is the original area multiplied by the determinant of the matrix of the linear transformation.

    Consider the circle of radius r = 1. The area of this circle is pi.
    Transform this circle into an ellipse of major and minor axes 2a and 2b. Then the area of this ellipse is pi det(M) = pi ab ......
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