A circle's center is at #(2 ,6 )# and it passes through #(3 ,1 )#. What is the length of an arc covering #(pi ) /3 # radians on the circle?

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Answer 1 · 2016-02-21T15:41:26

arc length = radius x arc angle
radius = #sqrt(26)#
arc angle = #(pi)/3#

arc length = #sqrt(26)((pi)/3) ~= 5.34#

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 (2,6), so #x_0=2 and y_0=6#
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+(1-6)^2=r^2#
#(1)^2+(-5)^2=r^2#
#1+25=r^2=26#
Therefore: r = #sqrt(26)#

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

arc angle = #(pi)/3#

arc length = #sqrt(26)((pi)/3) ~= 5.34#