I am trying to find the distance between the point $(p, 4p)$ and the parabola $y^{2} = 2px$, where $p$ is a fixed positive parameter.
So far, I have got $\varphi=\left(x-p\right)^{2}+\left(y-4p\right)^{2}+\lambda\left(y^{2}-2px\right)$
and the Lagrange equations \begin{align} \varphi_{x}& =2\left(x-p\right)-2\lambda p\\ \varphi_{y}& =2\left(y-4p\right)+\lambda y\\ \varphi_{\lambda} &=y^{2}-2px \end{align}
I am then confused as to whether I do $\varphi_{p}$ or am I treating $p$ as a constant? After this I am not sure of the next steps in the method to find an answer as to the distance between the point and the parabola.
Thanks