1. The problem is to understand the equation of a circle and analyze the given circle centered at (4, -7) with radius 5. 2. The general formula for a circle with center $(h, k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. For the given circle, the center is $(4, -7)$ and the radius is $5$. Substituting these values into the formula gives: $$ (x - 4)^2 + (y + 7)^2 = 5^2 $$ $$ (x - 4)^2 + (y + 7)^2 = 25 $$ 4. The point $(0, -7)$ lies on the circumference. To verify, substitute $x=0$ and $y=-7$ into the equation: $$ (0 - 4)^2 + (-7 + 7)^2 = (-4)^2 + 0^2 = 16 + 0 = 16 $$ Since $16 \neq 25$, the point $(0, -7)$ is not on the circle centered at $(4, -7)$ with radius 5. 5. However, the original equation $x^2 + y^2 = 25$ represents a circle centered at the origin $(0,0)$ with radius 5. 6. The point $(0, -7)$ is outside this circle because: $$ 0^2 + (-7)^2 = 49 > 25 $$ 7. To summarize, the equation $x^2 + y^2 = 25$ is a circle centered at $(0,0)$ with radius 5. 8. The circle centered at $(4, -7)$ with radius 5 is given by: $$ (x - 4)^2 + (y + 7)^2 = 25 $$ 9. The point $(0, -7)$ lies on the circle centered at $(4, -7)$ with radius 4, since: $$ (0 - 4)^2 + (-7 + 7)^2 = 16 + 0 = 16 $$ which corresponds to radius $4$, not $5$. Final answer: The equation $x^2 + y^2 = 25$ is a circle centered at $(0,0)$ with radius 5. The circle centered at $(4, -7)$ with radius 5 is: $$ (x - 4)^2 + (y + 7)^2 = 25 $$