The curve C C\,C has equation
y=184(2x−k),x≠k2 y = \frac{18}{4(2x - k)}, \quad x \neq \frac{k}{2} y=4(2x−k)18,x=2kwhere k k\,k is a positive constant and k≠2k \neq 2k=2.
Find dydx\displaystyle \frac{dy}{dx}dxdy giving your answer in simplest form in terms of kkk.
The point P P\,P with x x\,x coordinate 1 lies on CCC. Given that the gradient of the curve at P P\,P is -9, find the two possible values of kkk.
Given also that k<2k < 2k<2, find the equation of the normal to C C\,C at PPP, writing your answer in the form ax+by+c=0ax + by + c = 0ax+by+c=0, where a,b a, b\,a,b and c c\,c are integers to be found.
Show, using algebraic integration, that
∫13184(2x−k) dx=λln(5) \int_{1}^{3} \frac{18}{4(2x - k)} \, dx = \lambda \ln(5) ∫134(2x−k)18dx=λln(5)where λ \lambda\,λ is a constant to be found.
