Let $(c_1,c_2)$ be a fixed point in $\mathbb{R}^2$. How to maximize $|x_1x_2-c_1c_2|$ subject to the condition that $(x_1-c_1)^2+(x_2-c_2)^2<1$. i.e. $$\sup_{(x_1,x_2) \in \mathbb{R}^2:(x_1-c_1)^2+(x_2-c_2)^2<1} |x_1x_2-c_1c_2|=?$$
Note : This is a generalization of the problem of maximizing $|x_1x_2|$ subject to the condition $x_1^2+x_2^2<1$. I'm stuck with it. Any help would be much appreciated.
The allowable region for $(x_1,x_2)$ is a unit circle centered on $(c_1,c_2)$. The optimal point will be one where the hyperbola $x_1x_2=k$ is tangent to the circle and $k$ will be the maximum. It will be on the circle, so we can write $x_1=c_1+\cos \theta, x_2=c_2+\sin \theta$. Then $x_1x_2-c_1c_2=c_1\sin \theta+c_2\cos \theta + \sin \theta \cos \theta$. If $(c_1,c_2)$ is in the first quadrant we want $\theta$ in the first quadrant as well. We can now take a derivative with respect to theta and set it to zero, getting $$c_1\cos \theta -c_2 \sin \theta +\cos^2 \theta - \sin^2 \theta=0$$ Now solve for $\theta$
