Chapter 9.4, Problem 9.109P

Using Mohr’s circle, prove that the expression $I_{x^{\prime }}{I}_{y^{\prime }}-I^2{}_{x^{\prime }{y}^{\prime }}$ is independent of the orientation of the x′ and y′ axes, where $I_{x^{\prime }}$, $I_{y^{\prime }}$, and $I_{x^{\prime }{y}^{\prime }}$ represent the moments and product of inertia, respectively, of a given area with respect to a pair of rectangular axes x′ and y′ through a given point O. Also show that the given expression is equal to the square of the length of the tangent drawn from the origin of the coordinate system to Mohr’s circle.