Parameterize a line corresponding to t values

Question

Parameterize the line that passes through $P(2,-2)$ and $Q(6,2)$ so that the points $P$ and $Q$ correspond to the parameter values $t=11$ and $t=12$.

Answer 1

Let $x=a(t-11)+2,y=b(t-12)-2$ where $a,b$ are arbitrary constants

Now set $t=12$ to find $a,b$

Answer 2

The parametric equations of a line are in the form $$\left\{\begin{align}x&=x_1+t\Delta x\\ y&=y_1+t\Delta y,\end{align}\right.$$ where $(\Delta x,\Delta y)$ is the slope of the line and $(x_1,y_1)$ is a point located on the line.

Substituting $t=11$, we get $$\left\{\begin{align}2&=x_1+11\Delta x\\-2&=y_1+11\Delta y.\end{align}\right.$$

Substituting $t=12$, we get $$\left\{\begin{align}6&=x_1+12\Delta x\\2&=y_1+12\Delta y.\end{align}\right.$$

You now have two systems of two equations -- one involving $x_1$ and $\Delta x$, and the other involving $y_1$ and $\Delta y$. Can you take it from here?

Answer 3

Parametrization of the line PQ is given by $Qt+(1-t)P$ where $0\leq t\leq 1.$ Therefore parametrization of PQ is then $r(t)=(4t+2, 4t-2), 0\leq t\leq 1.$ Since you want to shift $t=0, t=1$ to $t=11, t=12$ respectively replace in $r(t)$ by $r(t-11)$ and required parametrization is $r^*(t)=(4t-42, 4t-46).$

Answer 4

Parameterizing a line means that you can write the co-ordinates of every point on the line in the form $x=at+b$ and $y=ct+d$. We have to find the values of $a, b, c, d$ from the co-ordinates of $P,Q$ and the values $t=11, 12$ at those points.

Looking at the $x$ co-ordinates of $P$ and $Q$ we see that

$+2=11a+b$
$+6=12a+b$

Subtracting the 1st equation from the 2nd we get $4=a$. Substituting this value into the 2nd equation we find $6=12*4+b=48+b$ so $b=-42$.

Then we do the same thing for the $y$ co-ordinates of $P, Q$ :

$-2=11c+d$
$+2=12c+d$

Subtracting the 1st equation from the 2nd again we get $4=c$ and substituting this value into the 2nd equation $+2=12*4+d=48+d$ so $d=-46$.

So the co-ordinates of any point on the line are

$x=4t-42=2(2t-21)$
$y=4t-46=2(2t-23)$

Check : When $t=11$ we get $x=2(22-21)=2$ and $y=2(22-23)=-2$ for $P$ and when $t=12$ we get $x=2(24-21)=6$ and $y=2(24-23)=2$ for $Q$.