Q. 54 - Calculus

Question

Show that the graph of the parametric equations

$x=a+(c-a)t,y=b+(d-b)t,t\in [0,1]$

is a line segment from $(a, b)$ to $(c, d).$

Step-by-Step Solution

Answer

The given information is proved.

Step 1: Given information

The graph of the parametric equations

$x=a+(c-a)t,y=b+(d-b)t,t\in [0,1]$

is a line segment from $(a, b)$ to 

Step 2: Calculation

Consider the two points $(a, b)(c, d)$.

The objective is to show that the line segment joining the two points is equal to the graph of the parametric equations.

If $\left(x_1,y_1\right)\left(x_2,y_2\right)$are any two points, then the equation for the line segment is given by $x=(1-t)x_1+t{x}_2,y=(1-t)y_1+t{y}_2$for some real number$t$.

Now write the line segment equations for the points $(a, b)(c, d)$.

Then,

$$\begin{gathered} x=(1-t)a+tc\left[since{x}_1=a,x_2=c\right] \\ x=a-ta+tc \end{gathered}$$

 By adjusting the terms

Now,

$$\begin{gathered} y=(1-t)b+td\left[\text{ since }{y}_1=b_{,}{y}_2=d\right] \\ y=b-tb+td \end{gathered}$$

$y=b+(d-b)t$ By adjusting the terms

Thus the line segment joining the points, $(a, b)(c, d)$is $x=a+(c-a) t, y=b+(d-b) t$which is equal to the given parametric equations.

Hence proved.