Let $A$ and $B$ be the matrices of two different conic sections. Then we call the conic sections associated to the matrices $\alpha A + \beta B$ (where $\alpha$ and $\beta$ can't be simultaneously equal to $0$) the family of conic sections defined by $A$ and $B$. In other words, the elements of the family of conic sections defined by the conic sections $ \mathbf{x}A\mathbf{x}^T = 0 $ and $ \mathbf{x}B\mathbf{x}^T = 0 $ are the conic sections with equation $ \mathbf{x}\left(\alpha A + \beta B\right)\mathbf{x}^T = 0 $, where $\alpha$ and $\beta$ can't be simultaneously equal to $0$.
Problem. Find a family of conic sections which contains exactly one circle!
I tried to approach this problem with arbitrary matrices, but I don't know how to use the condition that it has to be exactly one circle in it...
Can anyone help me with this?
The equation of a circle may not contain the mixed term $xy$. So for example
$$\alpha(x^2+y^2-1)+\beta xy=0.$$