Q. 58 - 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 $(-3, 5) to (1, 4)$

Step-by-Step Solution

Answer

The required parametric equations are $x=-3+4t,y=5-t,t\in [0,1]$

Step 1: Given Information

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

Step 2 : Calculation

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

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 $(-3,5)$ to $(1,4)$.

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

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

$$\begin{gathered} x=-3+(1+3)t \\ x=-3+4t \end{gathered}$$

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

Substitute the values in the equation $y=b+(d-b) t$, we get

$$\begin{gathered} y=5+(4-5)t\text{ since }a=-3,b=5,c=1,d=4 \\ y=5+(-1)t \\ y=5-t \end{gathered}$$

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

Therefore, the parametric equations are $x=-3+4t,y=5-t,t\in [0,1].$