Parametric Curves WEBASSIGN

Question

(a) Show that the parametric equations
x = x1 + (x2 − x1)t, y = y1 + (y2 − y1)t
where 0 ≤ t ≤ 1, describe the line segment that joins the points P1(x1, y1) and P2(x2, y2).

 

(b) Find parametric equations to represent the line segment from
(−2, 9) to (1, −1).
(Enter your answer as a comma-separated list of equations. Let x and y be in terms of t.)

Answer 1

I'm going to start by solving the first equation for t....which will give (x-x₁)/(x₂-x₁) =t.

 

You can now substitute this in the second equation to get....y=y₁ + (y₂-y₁)(x-x₁)/(x₂-x₁)

 

Rearranging, you get y-y₁ = (y₂-y₁) / (x₂-x₁) * (x-x₁) which is equivalent to y-y₁= m(x-x₁) which is the point-slope formula for a line.

 

So since t was varying between 0 and 1, 

plugging in t=0 you get x=x₁ and y = y₁ givin you the coordinate (x₁,y₁₎

similarly, plugging in t=1 leads to (x₂,y₂)

 

So this is a representation of a segment connecting these 2 points.

B) just plug in the points into your given formula from part a to get...

x=-2 +(1--2)t and y = 9 +(-1-9)t

or x=-2+3t , y=-9-10t

 

Hope this helped.