Question

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

Answer 1

Answer:

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

Step-by-step 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 $F(t)= (3cos(t)+4,3sin(t)+1)$.