We are given two circles, Circle A and Circle B, in the xy-plane with their equations. Here's how we can break it down:

Circle A:

The equation for Circle A is: $(x + 5)^2 + (y - 5)^2 = 4$ This is the standard form of a circle equation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• Center of Circle A: $(-5, 5)$
• Radius of Circle A: $\sqrt{4} = 2$

Circle B:

Circle B has the same center as Circle A but its radius is twice the radius of Circle A. Therefore, the radius of Circle B is: $2 \times 2 = 4$ The equation for Circle B is given as: $(x + 5)^2 + (y - 5)^2 = k$ Since the radius of Circle B is 4, and the equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, we can equate $r^2 = k$: $k = 4^2 = 16$

Thus, the value of $k$ is $16$.

Do you have any questions or would you like more details? Here are some related questions for further practice:

1. How would you write the equation of a circle with a center at $(3, -7)$ and a radius of 5?
2. If Circle A had a radius of 3 instead of 2, what would be the value of $k$ for Circle B?
3. What would the equation for Circle A be if its radius were halved?
4. How do you find the equation of a circle given three points on its circumference?
5. How does the equation of a circle change when it is translated horizontally by 2 units?

Tip: Remember that the general form of a circle's equation is directly tied to its center and radius. Always square the radius for the equation!