Question

22
score: 14/23 answered: 15/23
question 16
write the standard form of the equation of the circle having the given center and containing the given point.
center: (1,1); point: (1,2)

Explanation:

Step1: Recall the standard - form of a circle equation

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

Step2: Calculate the radius

The radius $r$ is the distance between the center $(1,1)$ and the point $(1,2)$ on the circle. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $x_1 = 1,y_1 = 1,x_2 = 1,y_2 = 2$. Then $r=\sqrt{(1 - 1)^2+(2 - 1)^2}=\sqrt{0 + 1}=1$.

Step3: Write the circle equation

Substitute $h = 1,k = 1,r = 1$ into the standard - form equation: $(x - 1)^2+(y - 1)^2=1^2$, which simplifies to $(x - 1)^2+(y - 1)^2=1$.

Answer:

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