Question

In the following exercises, determine whether the given points are collinear.

$(0,-5),(-2,-2),(2,-8)$

Short answer

The points $(0,-5),(-2,-2),(2,-8)$ are collinear.

Step-by-step solution

Step 1. Given

The points $(0,-5),(-2,-2),(2,-8)$

$(x_1,y_1)=(0,-5)$

$(x_2,y_2)=(-2,-2)$

$(x_3,y_3)=(2,-8)$

To find if the points are collinear.

Step 2. Test for collinear points

Three points$(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are collinear if and only if $\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Step 3. Substitute the points

Substitute the points $(x_1,y_1)=(0,-5)$,

$(x_2,y_2)=(-2,-2)$

$(x_3,y_3)=(2,-8)$

in the test for collinearity.

$\left|\begin{array}{ccc}0& -5& 1\\ -2& -2& 1\\ 2& -8& 1\end{array}\right|=0(-2+8)-(-5)(-2-2)+1(16+4)$

$=0+5(-4)+1\left(20\right)$

$=-20+20$

$=0$

The value of determinant is zero.

So the points are collinear.