Find the equation of the tangent line to the curve (a lemniscate) 2 ( x 2 + y 2 ) 2 = 25 ( x 2 − y 2 ) at the point (3,1). Write the equation of the tangent line in the form y = m x + b .Every time that we do it we get a ridiculous number for the slope (150/362)

Question

Find the equation of the tangent line to the curve (a lemniscate)   2  (      x    2    +      y    2        )    2    =  25  (      x    2    −      y    2    ) at the point (3,1). Write the equation of the tangent line in the form   y  =  m  x  +  b  .Every time that we do it we get a ridiculous number for the slope (150/362)

dkmc4175fl · Accepted Answer

We have   2  (      x    2    +      y    2        )    2    =  25  (      x    2    −      y    2    ).Since   (      x    2        )    ′    =  2  x     d  x, I would differentiate this as   2  (  2  (      x    2    +      y    2    )  (      x    2    +      y    2        )    ′    )  =  25  (  2  x     d  x  −  2  y     d  y  ) or   4  (      x    2    +      y    2    )  (  2  x     d  x  −  2  y     d  y  )  =  25  (  2  x     d  x  −  2  y     d  y  ) or   8  (  x  (      x    2    +      y    2    )  d  x  −  y  (      x    2    +      y    2    )  d  y  )  =  50  x     d  x  −  50  y     d  y or   (  8  x  (      x    2    +      y    2    )  −  50  x  )  d  x  =  (  8  y  (      x    2    +      y    2    )  −  50  y  )  d  y or                     d        y                    d        x              =                    8        x        (                  x          2                +                  y          2                )        −        50        x                    8        y        (                  x          2                +                  y          2                )        −        50        y            .Putting   x  =  3 and   y  =  1,                                                        d              y                                      d              x                                                  =                                            8              x              (                              x                2                            +                              y                2                            )              −              50              x                                      8              y              (                              x                2                            +                              y                2                            )              −              50              y                                                                =                                            24              (              9              +              1              )              −              150                                      8              (              8              +              1              )              −              50                                                                =                              90            22                                                    =                              45            11                              If the line is   y  =  m  x  +  b,   m  =  45      /    11 and   b  =  y  −  m  x  =  1  −  3  m  =  1  −  135      /    11  =  −  124      /    11All errors are (obviously) my fault, and I will accept all appropriate punishments.

Find the equation of the tangent line to the curve (a lemniscate) 2(x^2+y^2)^2=25(x^2−y^2) at the point (3,1). Write the equation of the tangent line in the form y=mx+b

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