For the circle x 2 + y 2 = r 2 , show that ∣ y ″ [ 1 + ( y ′ ) 2 ] 3 / 2 ∣Is x a constant as it disappear from the final equation? Should I only try to differentiate y and r?

Question

For the circle       x    2    +      y    2    =      r    2  , show that   ∣            y      ″              [      1      +      (              y        ′                    )        2                    ]                  3                      /                    2                      ∣Is   x a constant as it disappear from the final equation? Should I only try to differentiate   y and   r?

Adalyn Pitts · Accepted Answer

The first differentiation is straightforward,  x  +  y      y    ′    =  0.Then we will use a trick to speed up the computation:            (              1                  1          +                      y                          ′              2                                          )        ′    =  −                    y        ′                    y        ″                    (      1      +              y                  ′          2                            )                  3                      /                    2                      .But      1          1      +              y                  ′          2                      =      y                  x        2            +              y        2              =      y    r  immediately giving us  −                    y        ′                    y        ″                    (      1      +              y                  ′          2                            )                  3                      /                    2                      =            y      ′        r  from which the claim.

duandaTed05 · Accepted Answer

Hint:Compute the first derivative first, differentiating the circle equation:  (      x    2    +      y    2        )    ′    =  0    ⟺    2  x  +  2  y      y    ′    =  0    ⟺        y    ′    =  −      x    y    .We deduce, by the usual rules,      y    ″    =  −            (            x    y                      )              ′    =  −            1      ⋅      y      −      x              y        ′                    y      2        =  −            y      +                                                                                                  x                2                                                                                                                                                          y                                                                                y      2        =  −            r      2              y      3        .

For the circle x^2+y^2=r^2, show that ∣∣∣(y′′)/([1+(y′)^2]^(3/2))∣∣∣ =1/r Is x a constant as it disappear from the final equation? Should I only try to differentiate y and r?

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