The author gives sharp lower and upper bounds for the ratio of adjacency spectral radius and the clique number and the ratio of signless Laplacian spectral radius and the clique number, together with characterisation of extremal graphs. These results prove a conjecture from [\textit{M. Aouchiche}, Comparaison automatisée d'invariants en théorie des graphes. Montréal: École Polytechnique de Montréal (PhD Thesis) (2006)] and improve a result from [\textit{B. He} et al., Linear Algebra Appl. 438, No. 10, 3851--3861 (2013; Zbl 1282.05119)]. The paper also contains a sharp lower bound for the ratio of Laplacian spectral radius and the clique number.