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What are the equations of the tangents common to the circles x² + y²=1 , and (x-1)² + (y-3)² = 4?

I first used the T²=SS1 form which gives the joint equation of two tangents to a curve dream from an external point , but the equation of becoming too complex to solve ,please help me out . Thank you

KReiser
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  • https://math.stackexchange.com/questions/4057527/what-are-the-equations-of-the-tangents-common-to-the-circles-x2-y2-1-and – lab bhattacharjee Mar 11 '21 at 11:02
  • Looks like a popular question. – Gerry Myerson Mar 11 '21 at 11:11
  • The unit circle is self-dual, the other circle has dual $5y^2+6xy+6y-3x^2+2x+1=0.$ The four intersections of the duals give the four common tangents: $0x-y+1=0,x+0y+1=0,-\frac45 x+\frac35 y+1=0,-\frac35 x-\frac45 y +1=0.$ – Jan-Magnus Økland Mar 11 '21 at 11:21
  • See https://math.stackexchange.com/questions/211538/common-tangent-to-two-circles/4057624#4057624 – lab bhattacharjee Mar 11 '21 at 11:33
  • Thanks a lot !!! Jan-Magnus Økland , buy what is DUAL ??Please tell me . –  Mar 11 '21 at 14:54
  • Since you asked, the dual when the conic has matrix $M$ is the conic given by the inverse (or more generally the adjoint) matrix. The dual curve parametrizes the tangents to the curve, and common tangents therefore correspond to the intersection points of the duals. A point in the dual plane corresponds via the correspondence $x X_i+y Y_i+1=0,$ where $(X_i,Y_i)$ are the intersection points of the duals. – Jan-Magnus Økland Mar 11 '21 at 14:57
  • BTW if you @ me I get a notification. – Jan-Magnus Økland Mar 11 '21 at 14:58
  • Thanks @Jan-MagnusØkland , you are really helpful !!! –  Mar 11 '21 at 14:59

1 Answers1

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A quick sketch will show you that lines $x=-1$ and $y=1$ are two common tangents. The other two tangents are the reflections of these about the line through the centres.

Intelligenti pauca
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  • Thanks !!!!!!! I appreciate it , it's such a intuitive and quick way of solving this problem !! –  Mar 11 '21 at 14:26