Intersection point of tangents of a curve.

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 $m_2$.

• This is a model problem in differentiation. Please show us at least the result of the differentiation, anything more would be appreciated too. – астон вілла олоф мэллбэрг Nov 19 '16 at 7:48

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.