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

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First, we need to find the radius of the circle. We can use the distance formula to find the distance between the center of the circle ((1, 3)) and the given point ((2, 1)), which will give us the radius.

[ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substituting the coordinates, we get:

[ \text{Radius} = \sqrt{(2 - 1)^2 + (1 - 3)^2} ] [ \text{Radius} = \sqrt{1^2 + (-2)^2} ] [ \text{Radius} = \sqrt{1 + 4} ] [ \text{Radius} = \sqrt{5} ]

Now that we have the radius, to find the length of the arc covering ( \frac{\pi}{4} ) radians, we use the formula:

[ \text{Arc Length} = \text{radius} \times \text{angle in radians} ]

[ \text{Arc Length} = \sqrt{5} \times \frac{\pi}{4} ]

[ \text{Arc Length} = \frac{\sqrt{5}\pi}{4} ]

Therefore, the length of the arc covering ( \frac{\pi}{4} ) radians on the circle is ( \frac{\sqrt{5}\pi}{4} ).

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