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?

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Answer 1 · 2016-02-18T22:04:44

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

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

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#