Two tangents of the curve $x^2+2y^2+xy+x+y=10$ of slope $m_1$ and $m_2$ and pass through $(5,1)$. Find the point of intersection of other two tangents of slope $m_1$ and $m_2$.
Hint. These four lines form a circumscribed parallelogram around your ellipse.
Is it true that the diagonals of this parallelogram goes through the centre of the ellipse?
What is the centre of this ellipse?
Note that parallelogram diagonals bisect each other. Therefore the point of intersection of other two tangents is the symmetric point of $(5,1)$ with respect to the centre of the ellipse.