Question

20. Which set of parametric equations over the interval 0 ≤ t ≤ 1 defines a line segment with initial point (–5, 3) and terminal point (1, –6)?
x(t) = –5 + t; y(t) = 3 – 6t
x(t) = –5 + 3t; y(t) = 1 – 6t
x(t) = –5 + 6t; y(t) = 3 – 9t
x(t) = –5 + 8t; y(t) = 1 – 7t

Answer 1

Given:

Initial point = (–5, 3)

Terminal point = (1, –6)

To find:

The set of parametric equations over the interval 0 ≤ t ≤ 1.

Solution:

The interval is 0 ≤ t ≤ 1, initial point is (–5, 3) and terminal point is (1, –6). It means,

$x(0)=-5,y(0)=3$

$x(1)=1,y(1)=-6$

Put t=1 in each parametric equation.

In option A,

$x(1)=-5+1=-4\neq 1$

In option B,

$x(1)=-5+3(1)=-5+3=-2\neq 1$

In option D,

$x(1)=-5+8(1)=-5+8=3\neq 1$

Therefore, options A, B and D are incorrect.

In option C,

$x(1)=-5+6(1)=-5+6=1$

$y(1)=3-9(1)=3-9=-6$

Put t=0 in x(t) and y(t).

$x(0)=-5+6(0)=-5$

$y(0)=3-9(0)=3$

Since, only in option C $x(0)=-5,y(0)=3,x(1)=1,y(1)=-6$, therefore, the correct option is C.