Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 6B / b

The quadratic form has symmetric matrixwhose characteristic polynomial is $(\lambda -2)(\lambda -3)(\lambda -8)$. All its eigenvalues are positive, so $A$ is positive definite and ${\mathbf{x}}^{T}A\mathbf{x}=1$ is an ellipsoid. The spectral theorem for real symmetric matrices gives semi-axis lengthsThe nearest points lie in the eigendirection of the largest eigenvalue. Since $(5,2,1{)}^T$ is an $8$-eigenvector, they are