Question

Find parametric equations which describe each curve. No particular orientation is required.

a. The line containing the points (1, 2) and (-3,8).
b. The circle with radius 3 and center (4,1).

Answer 1

The equation of the line is
`(x(t),y(t))=(1-4t,2+6t)` and the equation of the circle is
`F(t)= (3cos(t)+4,3sin(t)+1)`.

Explanation:

(a) Given: The given points are
`(1,2)` and
`(-3,8)`.

To find: The parametric equation of line containing points
`(1,2)` and
`(-3,8)`.

We know that the parametric equation of line containing
`(x_(1) ,y_(1) )` and
`(x_(2) ,y_(2) )` is given by
`(x(t),y(t))=(x_(1)+(x_(2)-x_(1))t,y_(1)+(y_(2)-y_(1))t)` where
`t`∈
`[0,1]`.

Now,
`x(t)=1+(-3-1)t`

i.e,
`x(t)=1-4t`

And,
`y(t)=2+(8-2)t`

i.e,
`y(t)=2+6t`

Hence, the required parametric equation of the line is
`(x(t),y(t))=(1-4t,2+6t)`.

(b) Given: The radius of circle is 3 and centre is
`(4,1)`.

To find: The parametric equation of circle with radius 3 and centre
`(4,1)`.

We know that parametric equation of circle with radius
`r` and centre
`(h,k)` is given by
`F(t)= (x(t),y(t))` where
`x(t)=rcos(t)+h` and
`y(t)=rsin(t)+k`.

Now,
`x(t)=3cos(t)+4`

`y(t)=3sin(t)+1`

So, the parametric equation of circle having radius 3 and centre
`(h,k)` is
`F(t)= (3cos(t)+4,3sin(t)+1)`.

Hence, the required equation of the circle is
`F(t)= (3cos(t)+4,3sin(t)+1)`.