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How could I do something like this in TikZ?

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    Please add a minimal working example (MWE) that illustrates your problem. I.e. which part are you having difficulties with? – zeroth Apr 25 '13 at 9:23
  • Define a circle, construct the tangent point via the tangent coordinate system, transform the circle to an ellipse and apply the same transformation to the tangent point, draw the ellipse and the tangents. – Toscho Apr 25 '13 at 10:45

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