Q. 59 - 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. 

$From (\pi , 3) to (\pi , 8)$

Step-by-Step Solution

Answer

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

Step 1: Given information

The given pairs of points in the direction indicated from$(\pi , 3)$ to$(\pi , 8)$

Step 2: Simplification

Consider the points to $(\pi ,8)$.

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 $(\pi ,3)$to $(\pi ,8).$

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

$x=\pi +(\pi -\pi )tsincea=\pi ,b=3,c=\pi ,d=8$

$$\begin{gathered} x=\pi +\left(0\right)t \\ x=\pi \end{gathered}$$

Now take the points$(\pi ,3)$ to$(\pi ,8)$.

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

$$\begin{gathered} y=3+(8-3)t\text{ since }a=\pi ,b=3,c=\pi ,d=8 \\ y=3+\left(5\right)t \\ y=3+5t \end{gathered}$$

Thus, the parametric equations tor the line segment joining the pair of points$(\pi ,3),(\pi ,8)$ are$x=\pi ,y=3+5t,t\in [0,1]$

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