1. **State the problem:** We have a circle with equation $$x^2 + y^2 - 6x - 4y - 87 = 0$$ and two distinct points A and B on this circle. Point P moves such that $$AP = BP$$. We need to find the constant $$k$$ in the locus equation $$x + 2y + k = 0$$ of point P. 2. **Rewrite the circle equation in standard form:** Complete the square for $$x$$ and $$y$$: $$x^2 - 6x + y^2 - 4y = 87$$ $$x^2 - 6x + 9 + y^2 - 4y + 4 = 87 + 9 + 4$$ $$ (x - 3)^2 + (y - 2)^2 = 100$$ So, the circle has center $$C(3,2)$$ and radius $$10$$. 3. **Understand the locus condition:** The locus of points $$P$$ such that $$AP = BP$$ is the perpendicular bisector of segment $$AB$$. 4. **Since A and B lie on the circle and are distinct, the midpoint $$M$$ of $$AB$$ lies on the circle's center line. The locus line is the perpendicular bisector of $$AB$$, which passes through the midpoint $$M$$. 5. **Given the locus line equation:** $$x + 2y + k = 0$$ This line is the perpendicular bisector of $$AB$$. 6. **Find the center of the circle $$C(3,2)$$ and check if it lies on the locus line:** Substitute $$x=3$$ and $$y=2$$: $$3 + 2(2) + k = 0$$ $$3 + 4 + k = 0$$ $$k = -7$$ 7. **Conclusion:** The constant $$k$$ is $$-7$$. **Final answer:** $$k = -7$$