1. **Problem Statement:** Given point $A(-1,-5)$ and its reflection image $A'(3,-3)$, find the equation of the mirror line in the form $ax + by + c = 0$. 2. **Reflection Properties:** The mirror line is the perpendicular bisector of the segment joining $A$ and $A'$. This means: - It passes through the midpoint of $A$ and $A'$. - It is perpendicular to the line segment $AA'$. 3. **Find the midpoint $M$ of $AA'$:** $$M = \left(\frac{-1+3}{2}, \frac{-5 + (-3)}{2}\right) = (1, -4)$$ 4. **Find the slope of $AA'$:** $$m_{AA'} = \frac{-3 - (-5)}{3 - (-1)} = \frac{2}{4} = \frac{1}{2}$$ 5. **Slope of the mirror line:** Since the mirror line is perpendicular to $AA'$, its slope $m$ satisfies: $$m \times m_{AA'} = -1 \implies m = -2$$ 6. **Equation of the mirror line:** Using point-slope form with point $M(1,-4)$: $$y - (-4) = -2(x - 1)$$ $$y + 4 = -2x + 2$$ $$y = -2x - 2$$ 7. **Rewrite in standard form $ax + by + c = 0$:** $$2x + y + 2 = 0$$ **Final answer:** The equation of the mirror line is $$2x + y + 2 = 0$$.