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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 $m_2$.

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

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    $\begingroup$ Excellent hint [+1], that alas cannot be useful if the OP doesn't draw the figure. This is a general issue: an increasing number of maths students, faced to geometric problems do not draw figures, or when they do it, are not able to do the "way and back" between algebraic treatment and geometric viewing/intuition... $\endgroup$ – Jean Marie Nov 19 '16 at 9:31
  • $\begingroup$ I have fully understood the hints. Thanx. $\endgroup$ – Babai Nov 27 '16 at 11:19

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