Given a parametric function x = x o + v x t + 1 2 a x t 2 y = y o + v y t + 1 2 a y t 2 and point P at (c,d) , find all point(s) on the function that are distance R from point P, assuming that R &gt; 0 and none of the constants equal 0.

Question

Given a parametric function                    x                            =                  x          o                +                  v          x                t        +                  1          2                          a          x                          t          2                                    y                            =                  y          o                +                  v          y                t        +                  1          2                          a          y                          t          2                    and point P at (c,d) , find all point(s) on the function that are distance R from point P, assuming that   R  &amp;gt;  0 and none of the constants equal 0.

mallol3i · Accepted Answer

Step 1I&#039;m assuming that we&#039;re given   R  ,  c  ,  d  ,      x    0    ,      v    x    ,      a    x    ,      y    0    ,      v    y    ,      a    y    .The squared distance between   P  =  (  c  ,  d  ) and   (  x  ,  y  ) is given by1)                     (1)                              R          2                =        (        x        −        c                  )          2                +        (        y        −        d                  )          2                    Substitute the definition of2)   x  =      x          0        +      v          x        ×  t  +      1    2    ×      a          x        ×      t          2          y  =      y          0        +      v          y        ×  t  +      1    2    ×      a          y        ×      t          2      in (1)You&#039;re left with a polynomial in t. Solve for t, this will give you different values of   t  =  {      t    1    ,  …  ,      t    4    }  . . Substitute each       t    i   in (2) to get different (x,y) points.Step 2I&#039;m too lazy to LaTeX the following equation:  (  0.25      a          y              2        +  0.25      a          x              2        )      t          4        +  (  1.0      v          y               a          y        +  1.0      v          x               a          x        )      t          3          +  (      v          x              2        +  1.0      x          o               a          x        −  1.0      a          x           c  −  1.0      a          y           c  +      v          y              2        +  1.0      y          o               a          y        )      t          2          +  (  −  2.0      v          y           c  +  2.0      y          o               v          y        +  2.0      x          o               v          x        −  2.0      v          x           c  )  t  +      x          o              2        +  2.0      c          2        −  1.0      R          2          −  2.0      x          o           c  −  2.0      y          o              2        =  0Anyways, that&#039;s the degree 4 polynomial in t. To find a closed-form expressions for the roots  {      t    1    =  ⋯  ,      t    2    =  ⋯  ,      t    3    =  ⋯  ,      t    4    =  ⋯  }  ,you will need to do a lot of algebra on this polynomial. For example this.

Given a parametric function <mtable columnalign="right left right left right left right left rig

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