Find the value

If the points $(1,-5),(-4,5)$ and $(\lambda, 7)$ are collinear, then

$\left|\begin{array}{ccc}1 & -5 & 1 \\ -4 & 5 & 1 \\ \lambda & 7 & 1\end{array}\right|=0$

$\Rightarrow\left|\begin{array}{rcc}1 & -5 & 1 \\ -5 & 10 & 0 \\ \lambda & 7 & 1\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{2} \rightarrow R_{2}-R_{1}\right]$

$\Rightarrow\left|\begin{array}{ccc}1 & -5 & 1 \\ -5 & 10 & 0 \\ \lambda-1 & 12 & 0\end{array}\right|=0 \quad\left[\right.$ Applying $\left.R_{3} \rightarrow R_{3}-R_{1}\right]$

$\Rightarrow \Delta=\left|\begin{array}{cc}-5 & 10 \\ \lambda-1 & 12\end{array}\right|=0$

$\Rightarrow-60-10(\lambda-1)=0$

$\Rightarrow-60-10 \lambda+10=0$

$\Rightarrow-10 \lambda=50$

$\Rightarrow \lambda=-5$