A curve C is defined by the parametric equations: x = t 2 and y = t 3 - 3 t , how do you show that C has two tangents at the point (3,0) and find their equations?

Question

A curve C is defined by the parametric equations:       x    =          t      2       and       y    =          t      3        -    3    t  , how do you show that C has two tangents at the point (3,0) and find their equations?

Daniel Valdez · Accepted Answer

Step 1                    d        y                    d        x              =                                        d            y                                              d              t                                                                        d            x                                              d              t                                            =                  3                  t          2                -        3                    2        t                  x    =    3   at       t    =    ±          3       so there are two slopes for tangent lines            m      1        =                  3                              (                          3                        )                    2                -        3                    2                  3                      =          6              2                  3                      =          3      and            m      2        =                  3                              (            -                          3                        )                    2                -        3                    2                  (          -                      3                    )                      =          6              -        2                  3                      =    -          3      The equations are found by finding the equations of the lines through             (      3      ,      0      )       with slopes             3       and       -          3            y    =          3              (      x      -      3      )       and       y    =    -          3              (      x      -      3      )

A curve C is defined by the parametric equations: <mstyle displaystyle="true"> x =

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