Question

A circle's center is at `(3 ,4 )` and it passes through `(0 ,2 )`. What is the length of an arc covering `( pi ) /6` radians on the circle?

Answer 1

Center of circle is at `(3,4)`, Circle passes through `(0,2)`
Angle made by arc on the circle=`pi/6`, Length of arc`=??`

Let `C=(3,4)`, `P=(0,2)`

Calculating distance between `C` and `P` will giveus the radius of the circle.

`|CP|=sqrt((0-3)^2+(2-4)^2)=sqrt(9+4)=sqrt13`

Let the radius be denoted by `r`, the angle subtended by the arc at the center be denoted by `theta` and the length of the arc be denoted by `s`.

Then `r=sqrt13` and `theta=pi/6`

We know that:
`s=rtheta`

`implies s=sqrt13*pi/6=3.605/6*pi=0.6008pi`

`implies s=0.6008pi`

Hence, the length of arc is `0.6008pi`.