Show that the sum of the x and y intercepts of any tangent line to the curve x 1 / 2 + y 1 / 2 = 4 is equal to 16So far I have found the derivative, − y x , but am having trouble as to how I would found the point of tangency as as that is what I would think you do next?

Question

Show that the sum of the x and y intercepts of any tangent line to the curve       x          1              /            2        +      y          1              /            2        =  4 is equal to 16So far I have found the derivative,   −            y              x      , but am having trouble as to how I would found the point of tangency as as that is what I would think you do next?

Isaac Molina · Accepted Answer

x                  1        2              +      y                  1        2              =  4  ⇒            d      y              d      x        =  −            y      x      Now, the equation of a straight line that passes through a point   (      x    1    ,      y    1    ) on the curve is given by                              y          0                −                  y          1                                    =        −                                            y              1                                      x              1                                      (        0        −                  x          1                )        ⇒                  y          0                =                  y          1                +                              x            1                                y            1                              The y intercept,       y    0   occurs at   x  =  0. Thus,                              y          0                −                  y          1                                    =        −                                            y              1                                      x              1                                      (        0        −                  x          1                )        ⇒                  y          0                =                  y          1                +                              x            1                                y            1                              Similarly, the   x intercept occurs at   y  =  0. Thus,                    0        −                  y          1                                    =        −                                            y              1                                      x              1                                      (                  x          0                −                  x          1                )        ⇒                  x          0                =                  x          1                +                              x            1                                y            1                              Adding the intercepts, we find                              x          0                +                  y          0                                    =                  x          1                +        2                              x            1                                y            1                          +                  y          1                                                  =        (                              x            1                          +                              y            1                                    )          2                                                  =        (        4                  )          2                                                  =        16            as was to be shown!

Show that the sum of the x and y intercepts of any tangent line to the curve x^(1/2)+y^(1/2)=4 is equal to 16

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