Question

2.2
score: 13/23 answered: 14/23
question 15
write the standard form of the equation of the circle having the given center and containing the given point.
center: (0,0); point: (2,2)
question help: written example

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.

Step2: Determine the center values

Given the center $(h,k)=(0,0)$, the equation becomes $x^{2}+y^{2}=r^{2}$.

Step3: Calculate the radius

The distance between the center $(0,0)$ and the point $(2,2)$ on the circle is the radius $r$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $r=\sqrt{(2 - 0)^2+(2 - 0)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Step4: Substitute the radius into the equation

Substituting $r = 2\sqrt{2}$ into $x^{2}+y^{2}=r^{2}$, we get $x^{2}+y^{2}=(2\sqrt{2})^{2}=8$.

Answer:

$x^{2}+y^{2}=8$