1. **Problem 3a:** Find the centre of the circle $C$ with equation $$x^2 + y^2 - 4x + 10y = k.$$ 2. **Step 1:** Rewrite the equation by completing the square for $x$ and $y$. Group $x$ and $y$ terms: $$x^2 - 4x + y^2 + 10y = k.$$ 3. **Step 2:** Complete the square for $x$: $$x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4.$$ 4. **Step 3:** Complete the square for $y$: $$y^2 + 10y = (y^2 + 10y + 25) - 25 = (y + 5)^2 - 25.$$ 5. **Step 4:** Substitute back: $$(x - 2)^2 - 4 + (y + 5)^2 - 25 = k \\ (x - 2)^2 + (y + 5)^2 = k + 29.$$ 6. **Step 5:** The centre of the circle is at the point where the squared terms are zero: $$\boxed{(2, -5)}.$$ 7. **Problem 3b:** State the range of possible values for $k$. 8. **Step 6:** Since the right side represents the radius squared, it must be non-negative: $$k + 29 \geq 0 \\ k \geq -29.$$ 9. **Answer:** The range of $k$ is $$\boxed{k \geq -29}.$$ 10. **Problem 4a:** Find the centre $M$ of the circle with equation $$x^2 + y^2 - 20x - 24y + 195 = 0.$$ 11. **Step 7:** Complete the square for $x$ and $y$: $$x^2 - 20x = (x - 10)^2 - 100,$$ $$y^2 - 24y = (y - 12)^2 - 144.$$ 12. **Step 8:** Substitute back: $$(x - 10)^2 - 100 + (y - 12)^2 - 144 + 195 = 0 \\ (x - 10)^2 + (y - 12)^2 = 49.$$ 13. **Step 9:** The centre $M$ is $$\boxed{(10, 12)}.$$ 14. **Problem 4b:** Find the radius of the circle $C$. 15. **Step 10:** Radius $r$ is the square root of the right side: $$r = \sqrt{49} = 7.$$ 16. **Answer:** $$\boxed{7}.$$ 17. **Problem 4c:** Find the length of line $MN$, where $N = (25, 32)$. 18. **Step 11:** Use distance formula: $$MN = \sqrt{(25 - 10)^2 + (32 - 12)^2} = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25.$$ 19. **Answer:** $$\boxed{25}.$$ 20. **Problem 4d:** Find length $NP$ where $P$ is the point of tangency on circle $C$ and tangent passes through $N$. 21. **Step 12:** Length $NP$ is the length of the tangent from $N$ to circle $C$. Use formula for tangent length from external point: $$NP = \sqrt{(distance\ MN)^2 - r^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24.$$ 22. **Answer:** $$\boxed{24}.$$