Question

The coordinates of 3 points are a(-1, -6), b(3, -12) and c(k, 6). Find the value of k if a, b and c are collinear.

Plz answer quickly and step by step.

Answer 1

The value of
`k` is -9.

Explanation:

If
`A(x,y) = (-1,-6)`,
`B(x,y) =(3, -12)` and
`C(x,y) = (k,6)`, then
`\overrightarrow{BC} = \alpha\cdot \overrightarrow{BA}`. By vectors sum, we find each vector below:

`\overrightarrow{BA} = A(x,y)-B(x,y)` (1)

`\overrightarrow{BA} = (-1,-6)-(3,-12)`

`\overrightarrow{BA} = (-4, 6)`

`\overrightarrow{BC} = C(x,y)-B(x,y)` (2)

`\overrightarrow{BC} = (k,6)-(3,-12)`

`\overrightarrow{BC} = (k-3, 18)`

By substituting in the equation defined at the begining of this answer:

`(k-3, 18) =\alpha\cdot (-4, 6)`

`(k-3, 18) = (-4\cdot \alpha, 6\cdot \alpha)` (3)

The value of
`\alpha` is:

`6\cdot \alpha = 18` (3a)

`\alpha = 3`

If we know that
`\alpha = 3`, then the value of
`k` is:

`k-3 = -4\cdot \alpha`

`k = 3-4\cdot \alpha`

`k = 3-4\cdot (3)`

`k = 3-12`

`k = -9`

Then, the value of
`k` is -9.