1. **Problem Statement:** Find the coordinates of the limiting points of the coaxial system of circles given by the equations: $$x^2 + y^2 = 2x + 8y + 11$$ and $$x^2 + y^2 + 4x + 2y + 5 = 0$$ 2. **Rewrite the equations in standard form:** For the first circle: $$x^2 - 2x + y^2 - 8y = -11$$ Complete the square: $$x^2 - 2x + 1 + y^2 - 8y + 16 = -11 + 1 + 16$$ $$ (x - 1)^2 + (y - 4)^2 = 6$$ Radius $r_1 = \sqrt{6}$, center $C_1 = (1,4)$. For the second circle: $$x^2 + 4x + y^2 + 2y = -5$$ Complete the square: $$x^2 + 4x + 4 + y^2 + 2y + 1 = -5 + 4 + 1$$ $$ (x + 2)^2 + (y + 1)^2 = 0$$ Radius $r_2 = 0$, center $C_2 = (-2,-1)$ (a point circle). 3. **Coaxial system limiting points:** Limiting points lie on the line joining the centers and satisfy the power of point condition: The limiting points $P$ satisfy: $$|PC_1|^2 - r_1^2 = |PC_2|^2 - r_2^2$$ Let $P = (x,y)$ lie on the line through $C_1$ and $C_2$. Vector from $C_2$ to $C_1$ is: $$\vec{d} = (1 - (-2), 4 - (-1)) = (3,5)$$ Parametrize $P$ as: $$P = C_2 + t\vec{d} = (-2 + 3t, -1 + 5t)$$ 4. **Apply the power of point condition:** $$|P - C_1|^2 - r_1^2 = |P - C_2|^2 - r_2^2$$ Calculate: $$|P - C_1|^2 = (x - 1)^2 + (y - 4)^2$$ $$|P - C_2|^2 = (x + 2)^2 + (y + 1)^2$$ Substitute $P$: $$(( -2 + 3t ) - 1)^2 + (( -1 + 5t ) - 4)^2 - 6 = (( -2 + 3t ) + 2)^2 + (( -1 + 5t ) + 1)^2 - 0$$ Simplify: Left side: $$( -3 + 3t )^2 + ( -5 + 5t )^2 - 6 = (3t - 3)^2 + (5t - 5)^2 - 6$$ Right side: $$(3t)^2 + (5t)^2 = 9t^2 + 25t^2 = 34t^2$$ So: $$(3t - 3)^2 + (5t - 5)^2 - 6 = 34t^2$$ Expand: $$(9t^2 - 18t + 9) + (25t^2 - 50t + 25) - 6 = 34t^2$$ Combine terms: $$9t^2 + 25t^2 - 18t - 50t + 9 + 25 - 6 = 34t^2$$ $$34t^2 - 68t + 28 = 34t^2$$ Subtract $34t^2$ from both sides: $$-68t + 28 = 0$$ Solve for $t$: $$-68t = -28$$ $$t = \frac{28}{68} = \frac{7}{17}$$ 5. **Find the limiting point coordinates:** $$x = -2 + 3 \times \frac{7}{17} = -2 + \frac{21}{17} = \frac{-34 + 21}{17} = \frac{-13}{17}$$ $$y = -1 + 5 \times \frac{7}{17} = -1 + \frac{35}{17} = \frac{-17 + 35}{17} = \frac{18}{17}$$ 6. **Final answer:** The limiting point of the coaxial system is: $$\boxed{\left( -\frac{13}{17}, \frac{18}{17} \right)}$$