Question

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:

A line segment with initial point (–5, 3) and terminal point (1, –6).

To find:

The set of parametric equations over the interval 0 ≤ t ≤ 1 which defines the given line segment.

Solution:

Initial point is (–5, 3). So,

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

Terminal point is (1, –6).

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

Check which of the given set of parametric equations satisfy $x(0)=-5,y(0)=3,x(1)=1,y(1)=-6$.

Put t=1 in each set of parametric equations.

In option A,

$y(1)=3-6(1)=3-6=-3\neq -6$

So, option A is incorrect.

In option B,

$y(1)=1-6(1)=1-6=-5\neq -6$

So, option B is incorrect.

In option C,

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

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

Put t=0, in this set of parametric equations.

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

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

So, option C is correct.

In option D,

$y(1)=1-7(1)=1-7=-3\neq -6$

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

So, option D is incorrect.