Questions: Choose the best answer. Given: A(2,1), B(0,5), C(-1,2), AB and BC, center point (1,3) and r= Find the equation of the circle. (x-1)^2+(y-3)^2=sqrt(5) (x-1)^2+(y-3)^2=5 (x+1)^2+(y+3)^2=5 (x+1)^2+(y+3)^2=sqrt(5)

Solution Steps

To find the equation of the circle, we need to use the given center point and radius. The general form of the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Given the center point \((1, 3)\), we need to calculate the radius \(r\) using the distance formula between the center and one of the points on the circle.

1. Use the distance formula to calculate the radius \(r\) from the center \((1, 3)\) to one of the given points, say \(A(2, 1)\).
2. Substitute the center \((1, 3)\) and the calculated radius \(r\) into the circle equation \((x-1)^2 + (y-3)^2 = r^2\).
3. Compare the resulting equation with the given options to find the best answer.

To find the radius \( r \) of the circle, we use the distance formula between the center point \( (1, 3) \) and point \( A(2, 1) \):

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

The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \( h = 1 \), \( k = 3 \), and \( r^2 = 5 \):

\[ (x - 1)^2 + (y - 3)^2 = 5 \]

Now we compare the derived equation with the provided options:

1. \((x-1)^{2}+(y-3)^{2}=\sqrt{5}\)
2. \((x-1)^{2}+(y-3)^{2}=5\)
3. \((x+1)^{2}+(y+3)^{2}=5\)
4. \((x+1)^{2}+(y+3)^{2}=\sqrt{5}\)

The correct equation is \((x-1)^{2}+(y-3)^{2}=5\).

Final Answer