A risk-sensitive stochastic control problem with finite/infinite horizon is studied with a 1-dimensional controlled process defined by a linear SDE with a linear control-term in the drift. In the criterion function, a non-linear/quadratic term is introduced by using the solution to a Riccati differential equation, and hence, the problem is not ELQG (Exponential Linear Quadratic Gaussian) in general. For the problem, optimal value and control are calculated in explicit forms and the set of admissible risk-sensitive parameters is given in a concrete form. As applications, two types of large deviations control problems, i.e., maximizing an upside large deviations probability and minimizing a downside large deviations probability, are mentioned.