If `(x-a)^2+(y-b)^2=c^2`, for some `c > 0`, prove that `([1+((dy)/(dx))^2]^(3/2))/((d^2y)/(dx^2))`is a constant independent of a and b.

Explanation

`" We have "(x-a)^(2)+(y-b)^(2)=c^(2),cgt0`
Differentiating w.r.t. x, we get
`2(x-a)+2(y-b)y'=0`
`"or "(x-a)+(y-b)y'=0`
`"or "y'(x-a)/(y-b)`
Differentiating (1), w.r.t. x again, we get
`1+(y')^(2)+(y-b)y''=0`
`therefore" "([1+((dy)/(dx))^(2)]^(3/2))/((d^(2)y)/(dx^(2)))=([1+(y')^(2)]^(3/2))/(([-1+(y')^(2)])/(y-b))`
`=-(y-b)[1+(y')^(2)]^(1/2)`
`=-(y-b)[1+((x-a)/(y-b))^(2)]^(1/2)`
`=-[(y-b)^(2)+(x-a)^(2)]^(1/2)`
`=-c`