Question

right triangle abc is located at a(-1, -2), b(-1, 1) and c(-5, 1) on a coordinate plane. what is the equation of a circle a with radius ac?

1. ((x + 1)^2+(y + 2)^2 = 9)
2. ((x + 5)^2+(y - 1)^2 = 16)
3. ((x + 2)^2+(y + 2)^2 = 25)
4. ((x + 5)^2+(y - 1)^2 = 25)

Explanation:

Step1: Recall circle - equation formula

The standard form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. The center of circle $A$ is $A(-1,-2)$, so $h=-1$ and $k = - 2$.

Step2: Calculate the radius $r = AC$

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $A(-1,-2)$ and $C(-5,1)$, so $r=\sqrt{(-5+1)^2+(1 + 2)^2}=\sqrt{(-4)^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step3: Substitute values into circle - equation

Substitute $h=-1$, $k=-2$, and $r = 5$ into the standard - form equation $(x - h)^2+(y - k)^2=r^2$. We get $(x+1)^2+(y + 2)^2=25$.

Answer:

$(x + 1)^2+(y + 2)^2=25$