Question

ABC are collinear points and A is at (4, -5) while B is at (-3, 2). What is the location of C if AC is four times AB?

Answer 1

Task. ABC are collinear points and A is at $(4,-5)$ while B is at $(-3,2)$. What is the location of C if AC is four times AB?

Solution. Let us write the parametric equation of the line $AB$. It is parallel to a vector $AB=(-3-4,2-(-5))=(-7,7)$, whence it has the following equation:

$x=4-7t,\qquad y=-5+7t.$

The point $B$ corresponds to $t_{B}=1$.

Let $t_{C}$ be the parameter corresponding to $C$ such that $AC=4AB$. Then either

$t_{C}=4,\qquad\Rightarrow\qquad C=(4-7*4,-5+7*4)=(-24,23)$

or

$t_{C}=-4,\qquad\Rightarrow\qquad C=(4-7*(-4),-5+7*(-4))=(32,-33).$

Answer. $C=(-24,23)$ or $C=(32,-33)$.