Question

circle d has its center at (4,3) and point e(-2,2) is on the circle. determine if point f(-2,8) is also on the circle. drag and drop the answers to the boxes to complete the sentences. the complete equation of circle d is substituting the coordinates of point f(-2,8) to the equation of circle d results in a statement, so point f on the circle.

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. Given the center of circle $D$ is $(4,3)$, so the equation of circle $D$ is $(x - 4)^2+(y - 3)^2=r^2$.

Step2: Calculate the radius using point $E(-2,2)$

The distance between the center $(4,3)$ and point $E(-2,2)$ is the radius $r$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $r=\sqrt{(-2 - 4)^2+(2 - 3)^2}=\sqrt{(-6)^2+(-1)^2}=\sqrt{36 + 1}=\sqrt{37}$. So the equation of circle $D$ is $(x - 4)^2+(y - 3)^2 = 37$.

Step3: Substitute the coordinates of point $F(-2,8)$ into the circle equation

Substitute $x=-2$ and $y = 8$ into $(x - 4)^2+(y - 3)^2$. We get $(-2 - 4)^2+(8 - 3)^2=(-6)^2+5^2=36 + 25=61$. Since $61
eq37$.

Answer:

The complete equation of circle $D$ is $(x - 4)^2+(y - 3)^2 = 37$. Substituting the coordinates of point $F(-2,8)$ to the equation of circle $D$ results in a false statement, so point $F$ is not on the circle.