Question

the image shows a circle with center (4,6) and radius 10 units.
select all points that lie on the circle.
a (-4,12)
b (-6,6)
c (4,6)
d (13,10)
e (4,16)

Explanation:

Step1: Recall circle equation

The standard equation of a circle with center $(h,k)$ and radius $r$ is $(x-h)^2+(y-k)^2=r^2$. For center $(4,6)$ and $r=10$, the equation is $(x-4)^2+(y-6)^2=100$.

Step2: Test point A $(-4,12)$

Substitute $x=-4,y=12$:
$$(-4-4)^2+(12-6)^2=(-8)^2+6^2=64+36=100$$
This equals $100$, so the point lies on the circle.

Step3: Test point B $(-6,6)$

Substitute $x=-6,y=6$:
$$(-6-4)^2+(6-6)^2=(-10)^2+0^2=100+0=100$$
This equals $100$, so the point lies on the circle.

Step4: Test point C $(4,6)$

Substitute $x=4,y=6$:
$$(4-4)^2+(6-6)^2=0^2+0^2=0$$
This does not equal $100$, so the point is the center, not on the circle.

Step5: Test point D $(13,10)$

Substitute $x=13,y=10$:
$$(13-4)^2+(10-6)^2=9^2+4^2=81+16=97$$
This does not equal $100$, so the point is not on the circle.

Step6: Test point E $(4,16)$

Substitute $x=4,y=16$:
$$(4-4)^2+(16-6)^2=0^2+10^2=0+100=100$$
This equals $100$, so the point lies on the circle.

Answer:

A. $(-4,12)$, B. $(-6,6)$, E. $(4,16)$