1. **Stating the problem:** We have two circles: - Circle 1 with center $A(-1,-3)$ passing through point $P(-2,4)$. - Circle 2 with center $B(-8,6)$ passing through point $Q(-6,3)$. We need to find the equations of these circles and determine their mutual position. 2. **Formula for circle equation:** The equation of a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Find radius of Circle 1:** Radius $r_1$ is the distance between $A(-1,-3)$ and $P(-2,4)$: $$ r_1 = \sqrt{(-2 + 1)^2 + (4 + 3)^2} = \sqrt{(-1)^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} $$ 4. **Equation of Circle 1:** $$ (x + 1)^2 + (y + 3)^2 = 50 $$ 5. **Find radius of Circle 2:** Radius $r_2$ is the distance between $B(-8,6)$ and $Q(-6,3)$: $$ r_2 = \sqrt{(-6 + 8)^2 + (3 - 6)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} $$ 6. **Equation of Circle 2:** $$ (x + 8)^2 + (y - 6)^2 = 13 $$ 7. **Determine mutual position:** Calculate distance $d$ between centers $A$ and $B$: $$ d = \sqrt{(-8 + 1)^2 + (6 + 3)^2} = \sqrt{(-7)^2 + 9^2} = \sqrt{49 + 81} = \sqrt{130} $$ 8. **Compare $d$ with $r_1 + r_2$ and $|r_1 - r_2|$:** $$ r_1 + r_2 = \sqrt{50} + \sqrt{13} \approx 7.07 + 3.61 = 10.68 $$ $$ |r_1 - r_2| = |\sqrt{50} - \sqrt{13}| \approx |7.07 - 3.61| = 3.46 $$ Since: $$ |r_1 - r_2| < d < r_1 + r_2 $$ or numerically: $$ 3.46 < 11.40 < 10.68 $$ This is false because $d \approx 11.40$ is greater than $r_1 + r_2 \approx 10.68$. Therefore, the circles are **separate and do not intersect**. **Final answer:** - Circle 1: $$(x + 1)^2 + (y + 3)^2 = 50$$ - Circle 2: $$(x + 8)^2 + (y - 6)^2 = 13$$ - Mutual position: The circles are separate (no common points).