Question

A circle's center is at `(2 ,4 )` and it passes through `(3 ,1 )`. What is the length of an arc covering `(13pi ) /12` radians on the circle?

Answer 1

arc length = radius x arc angle
radius = `sqrt(10)`
arc angle = `(13pi)/12`

arc length = `sqrt(10)((13pi)/12) ~= 10.76`

Explanation:

First find the radius, r, using the equation of a circle:
`(x-x_0)^2+(y-y_0)^2=r^2`

The circle's centre is (2,4), so `x_0=2 and y_0=4`
The circle passes through (3,1), so at this point x=3 and y=1.

Inputting all this into the circle equation:
`(3-2)^2+(1-4)^2=r^2`
`(1)^2+(-3)^2=r^2`
`1+9=r^2=10`
Therefore: r = `sqrt(10)`

Now we can find the arc length using:
arc length = radius x arc angle

arc angle = `(13pi)/12`

arc length = `sqrt(10)((13pi)/12) ~= 10.76`