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To solve this problem numerically, we are looking for a surface that is orthogonal to the given one-parameter family of surfaces: \[ z = cxy(x^2 + y^2) \] and also passes through the hyperbola: \[ x^2 - y^2 = a^2, \quad z = 0 \] ### Step 1: Finding the Gradient of the Given Surface The first step in finding the surface orthogonal to a _____ _________ ___ ____ ___ ___ ___.
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