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Show that the graph of the parametric equations

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

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

Short Answer

Expert verified

The given information is proved.

Step by step solution

01

Given information

The graph of the parametric equations

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

is a line segment from (a,b)to

02

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 x1,y1x2,y2are any two points, then the equation for the line segment is given by x=(1-t)x1+tx2,y=(1-t)y1+ty2for some real numbert.

Now write the line segment equations for the points localid="1653036356110" (a,b)(c,d).

Then,

localid="1653036358431" x=(1-t)a+tcsincex1=a,x2=cx=a-ta+tc

By adjusting the terms

Now,

localid="1653036360951" y=(1-t)b+tdsincey1=b,y2=dy=b-tb+td

localid="1653036364252" y=b+(d-b)tBy adjusting the terms

Thus the line segment joining the points, localid="1653036370197" (a,b)(c,d)is localid="1653036367612" x=a+(c-a)t,y=b+(d-b)twhich is equal to the given parametric equations.

Hence proved.

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