1. **State the problem:** We need to determine which points lie on the circle with center $O(2,3)$ and radius $4$. 2. **Formula:** The equation of a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ Here, $h=2$, $k=3$, and $r=4$, so: $$ (x - 2)^2 + (y - 3)^2 = 16 $$ 3. **Check each point:** Substitute each point into the equation and see if it satisfies it. - For $(1, -\frac{3}{4})$: $$ (1 - 2)^2 + \left(-\frac{3}{4} - 3\right)^2 = (-1)^2 + \left(-\frac{15}{4}\right)^2 = 1 + \frac{225}{16} = \frac{16}{16} + \frac{225}{16} = \frac{241}{16} = 15.0625 \neq 16 $$ - For $(-2, 3)$: $$ (-2 - 2)^2 + (3 - 3)^2 = (-4)^2 + 0^2 = 16 + 0 = 16 $$ This point lies on the circle. - For $(6, 5)$: $$ (6 - 2)^2 + (5 - 3)^2 = 4^2 + 2^2 = 16 + 4 = 20 \neq 16 $$ - For $(2, -1)$: $$ (2 - 2)^2 + (-1 - 3)^2 = 0^2 + (-4)^2 = 0 + 16 = 16 $$ This point lies on the circle. 4. **Answer:** The points on the circle are $(-2, 3)$ and $(2, -1)$.