1. **Problem Statement:** Given the circle equation $$(x + h)^2 + (y + k)^2 = r^2$$ with radius $r > 0$, determine which of the statements I. $hk < 0$, II. $h > -r$, and III. $k < r$ are true. 2. **Understanding the center:** The center of the circle is at $$(-h, -k)$$ because the equation is in the form $$(x - x_0)^2 + (y - y_0)^2 = r^2$$ where $x_0 = -h$ and $y_0 = -k$. 3. **Given graph description:** - The circle is centered in the first quadrant (positive $x$ and $y$). - This means $$-h > 0$$ and $$-k > 0$$. 4. **Analyze the signs of $h$ and $k$:** Since $$-h > 0$$, it follows that $$h < 0$$. Since $$-k > 0$$, it follows that $$k < 0$$. 5. **Check statement I: $hk < 0$** Since both $h$ and $k$ are negative, their product $$hk > 0$$ (negative times negative is positive). Therefore, statement I is **false**. 6. **Check statement II: $h > -r$** Since $h < 0$ and $r > 0$, consider the inequality: $$h > -r$$ Because $h$ is negative but greater than $-r$ means $h$ is to the right of $-r$ on the number line. Since the center $(-h, -k)$ is in the first quadrant, $-h > 0$ so $h < 0$ but must be greater than $-r$ (which is negative). This is plausible and generally true because the center must be within $r$ units from the origin. So statement II is **true**. 7. **Check statement III: $k < r$** Since $k < 0$ and $r > 0$, $k$ is definitely less than $r$. So statement III is **true**. 8. **Conclusion:** Only statements II and III are true. **Final answer:** C. II and III only