<s>[INST] Find an unknown coefficient in a line equation... So, I have to find the unknown coefficient in this line: $$x+y+C=0$$ so that it is a tangent to this circle: $$x^2+y^2-5x-7y+6=0$$
I've transformed the circle equation to this form:
$$(x-\frac{5}{2})^2+(y-\frac{7}{2})^2=\frac{25}{2}$$
But I can't see how it can help me lol :D
 [/INST] Substitute $y = C-x$ into the equation of the circle to get: 
$x^2+(C-x)^2-5x-7(C-x)+6 = 0$
$2x^2+(2-2C)x+(C^2-7C+6) = 0$
If the line is tangent to the circle, this equation has only one solution for $x$. So the discriminant of the quadratic should be $0$, i.e. 
$(2-2C)^2 - 4 \cdot 2 \cdot (C^2-7C+6) = 0$
This will give you two solutions for $C$. 
($C=1$ and $C=11$)
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