Question

A circle has a center at `(3 ,2 )` and passes through `(1 ,1 )`. What is the length of an arc covering `(3pi ) /4` radians on the circle?

Answer 1

`color(blue)((3pisqrt(5))/4)`

Explanation:

If the centre of the circle has coordinates `(3,2)` and the point `(1,1)` lies on the circumference, then the length of the radius is the distance between these points. This can be found using the distance formula.

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`d=sqrt((3-1)^2+(2-1)^2)=sqrt(5)`

If we rotate the terminal side `(3pi)/4` to form a sector, then the angle subtended by the arc at the centre will also be `(3pi)/4`

Arc length is given by:

`rtheta`

`:.`

`sqrt(5)((3pi)/4)=(3pisqrt(5))/4`