Find the value of $p$ for which the points $(-5,1)$, $(1,p)$ and $(4,-2)$ are collinear.

The correct option is A. $-1$
The given points are $A(-5,1)$, $B(1,p)$ and $C(4,-2)$

We have $(x_1=-5,y_1=1),(x_2=1,y_2=p)$ and $(x_3=4,y_3=-2)$

The given points $A,B$ and $C$ are collinear.

Therefore, slope of $AB=$ slope of $BC$.

$\Rightarrow \frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_2}$

$\Rightarrow \frac{p-1}{1-(-5)}=\frac{-2-p}{4-1}$

$\Rightarrow \frac{p-1}{1+5}=\frac{-2-p}{3}$

$\Rightarrow p-1=\frac{-2-p}{3}\times 6$

$\Rightarrow p-1=(-2-p)\times 2$

$\Rightarrow p-1=-4-2p$

$\Rightarrow p+2p=-4+1$

$\Rightarrow 3p=-3$

$\therefore p=-1$