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