Question

Find the equations of the planes that passes through $(1,1,-1),(6,4,-5),(-4,-2,3)$

Answer 1

The given points are $A(1,1,-1),B(6,4,-5)$ and $C(-4,-2,3)$.
$\left|\begin{array}{ccc}1& 1& -1\\ 6& 4& -5\\ -4& -2& 3\end{array}\right|=(12-10)-(18-20)-(12+16)=2+2-4=0$
$\Rightarrow$ Points are collinear.
Since $A,B,C$ are collinear points, there will be infinite number of planes passing through the given points.