<s>[INST] Determine conics by four points and a tangent line. I have small troubles determining conics that go through four points and have a given tangent line.
More specifically $P_1 = (0:1:0), P_2 = (0:0:1), P_3 = (1:0:1), P_4 = (1:-1:0) \in \mathbb{RP}^2$ and $ t: \{x-y+z=0\}$.
I can determine the conics as far as $ax^2+axy+2eyz-axz=0,\  a,e \in \mathbb{R}$ but don't really succeed with incorporating the information of the tangent line.
I read substituting in $z = -x +y$ should work, but i have no clue how to proceed with the obtained result $ax^2+ey^2-exy=0$. 
Help is appreciated a lot (also other ways of dealing with the tangent!).
 [/INST] $\det(\begin{pmatrix}x^2&xy&y^2&xz&yz&z^2\\0&0&1&0&0&0\\0&0&0&0&0&1\\1&0&0&1&0&1\\1&-1&1&0&0&0\\a^2&ab&b^2&ac&bc&c^2\end{pmatrix})=a(c-b-a)yz+bc(-xz+xy+x^2)=0$
Now assume $(a:b:c)$ is the point of tangency making $b=a+c$ and $y=x+z$, substitute and expand:
$2((ac+c^2)x^2-a^2xz-a^2z^2)$
The discriminant reduces to $a^2(2c+a)^2$, so $a=-2c,b=-c,c=c$, which makes the equation $-c^2(8yz-xz+xy+x^2)$, and this with $z=1$ is a hyperbola that actually is tangent to $y=x+1$ at $(-2,-1)$.
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