1. The problem statement, all variables and given/known data We have a surface z^2 = 4x^2 + 2yx + 5y^2 , find the shortest distance to Origin. 2. Relevant equations 3. The attempt at a solution My trouble is , i think z^2 - 4x^2 + 2yx + 4y^2 = 0 as a constraint to function L = x^2 + y^2 + z^2 (Square of distance formula. If distance is minimum , square of it should be too). Now i will use grad(L) = Lambda*grad(surface) But while finding the gradient of surface , i need derivatives respect to x , y and z as vector components. Now , while taking the derivative of constraint respect to x , should i use the implicit differentiaton ? My book just directly takes the differential of it respect to x , doesn't use the implicit diff. When dealing with plane equations , multivariable calculus , when to use implicit diff and when not to ? How could i understand if z is a function of x and y or they all are three variables ?