Integral of a Gaussian restricted to a plane

Question

Given $A\in\mathbb R^{N\times N}$ as a symmetric positive definite matrix, I want to show that $$\int_{\mathbb R^N}\delta(\theta-n^Tx)e^{-(x-x_0)^TA(x-x_0)/2}\,dx=\frac{e^{-(\theta-n^Tx_0)^2/2n^TA^{-1}n}}{\sqrt{(2\pi)^{1-N}(n^TA^{-1}n)\det A}}$$ I get that this is almost a surface integral of the Gaussian over a plane with $n$ as the normal vector and the plane is parametrized by $n^Tx=\theta$, but I'm having trouble with substituting the plane parametrization into the integral in a way that can be evaluated analytically.