Question

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

$(-2,1),(-4,4),(0,-2)$

Short answer

The points $(-2,1),(-4,4),(0,-2)$ are collinear

Step-by-step solution

Step 1. Given

The points $(-2,1),(-4,4),(0,-2)$

$(x_1,y_1)=(-2,1)$

$(x_2,y_2)=(-4,4)$

$(x_3,y_3)=(0,-2)$

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, $(x_1,y_1)=(-2,1)$,

$(x_2,y_2)=(-4,4)$d

$(x_3,y_3)=(0,-2)$

in the test for collinearity.

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

$=-2\left(6\right)-1(-4)+1\left(8\right)$

$=-12+12$

$=0$

The value of determinant is zero.

So the points are collinear.