Respuesta :
Answer:
The equation of the line is [tex](x(t),y(t))=(1-4t,2+6t)[/tex] and the equation of the circle is [tex]F(t)= (3cos(t)+4,3sin(t)+1)[/tex].
Step-by-step explanation:
(a) Given: The given points are [tex](1,2)[/tex] and [tex](-3,8)[/tex].
To find: The parametric equation of line containing points [tex](1,2)[/tex] and [tex](-3,8)[/tex].
We know that the parametric equation of line containing [tex](x_{1} ,y_{1} )[/tex] and [tex](x_{2} ,y_{2} )[/tex] is given by [tex](x(t),y(t))=(x_{1}+(x_{2}-x_{1})t,y_{1}+(y_{2}-y_{1})t)[/tex] where [tex]t[/tex]∈[tex][0,1][/tex].
Now, [tex]x(t)=1+(-3-1)t[/tex]
i.e, [tex]x(t)=1-4t[/tex]
And, [tex]y(t)=2+(8-2)t[/tex]
i.e, [tex]y(t)=2+6t[/tex]
Hence, the required parametric equation of the line is [tex](x(t),y(t))=(1-4t,2+6t)[/tex].
(b) Given: The radius of circle is 3 and centre is [tex](4,1)[/tex].
To find: The parametric equation of circle with radius 3 and centre [tex](4,1)[/tex].
We know that parametric equation of circle with radius [tex]r[/tex] and centre [tex](h,k)[/tex] is given by [tex]F(t)= (x(t),y(t))[/tex] where [tex]x(t)=rcos(t)+h[/tex] and [tex]y(t)=rsin(t)+k[/tex].
Now, [tex]x(t)=3cos(t)+4[/tex]
[tex]y(t)=3sin(t)+1[/tex]
So, the parametric equation of circle having radius 3 and centre [tex](h,k)[/tex] is [tex]F(t)= (3cos(t)+4,3sin(t)+1)[/tex].
Hence, the required equation of the circle is [tex]F(t)= (3cos(t)+4,3sin(t)+1)[/tex].
