Q. 57 - Calculus

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$(-3, 5)$

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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 $a=1, b=4, c=-3, d=5$

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

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 $x=1-4t,y=4+t,t\in [0,1]$.