Question

circle n has its center at (0, 4), and point m (2,0) is on the circle. prove or disprove that point o (4, 8) is also on the circle. drag and drop the answers to the boxes to complete the proof. the equation of circle n without yet knowing the radius is box. since point m is on the circle, we can substitute its coordinates into the equation of the circle to have box. then, we can find the radius of the circle to be box. so, the equation of the circle now becomes box. substituting now the coordinates of point o to the equation of the circle and simplifying it

Explanation:

Step1: Write general circle - equation

The general equation of a circle with center $(h,k)$ is $(x - h)^2+(y - k)^2=r^2$. Given the center $(0,4)$, the equation is $(x - 0)^2+(y - 4)^2=r^2$, or $x^{2}+(y - 4)^{2}=r^{2}$.

Step2: Substitute point M

Substitute $x = 2$ and $y = 0$ into the equation $x^{2}+(y - 4)^{2}=r^{2}$. We get $2^{2}+(0 - 4)^{2}=r^{2}$, which simplifies to $4+( - 4)^{2}=r^{2}$, then $4 + 16=r^{2}$, so $r^{2}=20$.

Step3: Write the complete - circle equation

The equation of the circle is $x^{2}+(y - 4)^{2}=20$.

Step4: Check point O

Substitute $x = 4$ and $y = 8$ into the equation $x^{2}+(y - 4)^{2}=20$. We have $4^{2}+(8 - 4)^{2}=16+16 = 32
eq20$.

Answer:

Point $O(4,8)$ is not on the circle.