Question

If (x − a)² + (y − b)² = c², for some c > 0, prove that `([1 + (dy/dx)^2]^(3/2))/((d^2y)/dx^2)` is a constant independent of a and b.

Answer 1

The given relation is (x − a)² + (y − b)² = c², c > 0.

Let x − a = c cos θ and y − b = c sin θ.

Therefore, `dx/(d theta) = -c sintheta  "And"  dy/(d theta) = c cos theta`

`therefore dy/dx = - cot theta`

Differentiate both sides with respect to θ, we get

`d/(d theta)(dy/dx) = d/(d theta) (- cot theta)`

Or, `d/dx(dy/dx)dx/(d theta) = cosec^2theta`

Or, `(d^2y)/(dx^2)(- c sin theta) = cosec^2theta`

`(d^2y)/(dx^2) = -(cosec^3 theta)/(c)`

`therefore ([1 + (dy/dx)^2]^(3/2))/((d^2y)/dx^2) = (c[1 + cot^2 theta]^(3/2))/(-cosec^3theta)`

= `(-c(cosec^2 theta)^(3/2))/(-cosec^3theta)`

= −c ...(Which is constant and is independent of a and b.)