Question

use the result of Exercise 54 to find parametric equations for the line segments connecting the given pairs of points in the direction indicated.

From $(1,4)$ to$(-3,5)$

Short answer

The parametric equations are $x=1-4t,y=4+t,t\in [0,1]$

Step-by-step solution

Given Information 

The points are from$(1,4)$ to $(-3,5)$

Calculation

Consider the points $(1,4)$to $(-3,5)$.

The objective is to find the parametric equations for the line segment joining the pair of points.

The formula for the line segment joining the pair of points $(a,b)$to $(c,d)$is as follows, $x=a+(c-a)t,y=b+(d-b)t,t\in [0,1]$.

Here to find the parametric equations substitute the given values in the equation of line segment.

Now take the points $(1,4)$to$(-3,5)$.

substitute the values in the equation $x=a+(c-a)t$we get,

$x=1+(-3-1)t$since $a=1,b=4,c=-3,d=5$

$$\begin{gathered} x=1+(-4)t \\ x=1-4t \end{gathered}$$

Now take the points $(1,4)$to $(-3,5)$.

Substitute the values in the equation$y=b+(d-b)t$, we get $y=4+(5-4)t$since $a=1,b=4,c=-3,d=5$

$$\begin{gathered} y=4+\left(1\right)t \\ y=4+t \end{gathered}$$

Thus, the parametric equations for the line segment joining the pair of points $(1,4),(-3,5)$are $x=1-4t,y=4+t,t\in [0,1].$

Therefore, the parametric equations are localid="1652370781798" $x=1-4t,y=4+t,t\in [0,1]$.