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:

The value of [tex]k[/tex] is -9.

Step-by-step explanation:

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

[tex]\overrightarrow{BA} = A(x,y)-B(x,y)[/tex] (1)

[tex]\overrightarrow{BA} = (-1,-6)-(3,-12)[/tex]

[tex]\overrightarrow{BA} = (-4, 6)[/tex]

[tex]\overrightarrow{BC} = C(x,y)-B(x,y)[/tex] (2)

[tex]\overrightarrow{BC} = (k,6)-(3,-12)[/tex]

[tex]\overrightarrow{BC} = (k-3, 18)[/tex]

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

[tex](k-3, 18) =\alpha\cdot (-4, 6)[/tex]

[tex](k-3, 18) = (-4\cdot \alpha, 6\cdot \alpha)[/tex] (3)

The value of [tex]\alpha[/tex] is:

[tex]6\cdot \alpha = 18[/tex] (3a)

[tex]\alpha = 3[/tex]

If we know that [tex]\alpha = 3[/tex], then the value of [tex]k[/tex] is:

[tex]k-3 = -4\cdot \alpha[/tex]

[tex]k = 3-4\cdot \alpha[/tex]

[tex]k = 3-4\cdot (3)[/tex]

[tex]k = 3-12[/tex]

[tex]k = -9[/tex]

Then, the value of [tex]k[/tex] is -9.

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