Related to problem 2.19 of convex optimization book by Stephen Boyd

Question

In the solution of problem 2.19 of the book it is mentioned that $$\{x|x^TQx+2q^Tx\leq r,c^Tx+d>0\}$$ where $Q=A^TP^{-1}A-cc^T$, $q=b^TP^{-1}A+dc$ and $r=d^2-b^TP^{-1}b$. It is written that if $Q$ is positive definite then the above set is the intersection of an ellipsoid and the half space. The ellipsoid part comes from the condition $x^TQx+2q^Tx\leq r$ which can be equalently written as $$(x+Q^{-1}q)^TQ(x+Q^{-1}q)\leq d^2-b^TP^{-1}b+q^{T}Q^{-1}q$$ ($P$ is positive definite)This equality matches with the inequality described for ellipsoid if the right side of the inequality is positive. However, I do not know how to prove that the right side is always positive. Any help in this regard will be much appreciated. Thanks in advance.