The circle C has equation $x^2 + y^2 - 20x - 24y + 195 = 0$ The centre of C is at the point M - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 6

From the completed square, the equation $(x - 10)^2 + (y - 12)^2 = 7$ shows that the radius $r$ can be found as follows:
$r = ext{sqrt}(7)$
Thus the radius of the circle C is $\sqrt{7}$.

We can use the distance formula to find the length of the line MN, where $M(10, 12)$ and $N(25, 32)$:
$MN = \sqrt{(25 - 10)^2 + (32 - 12)^2}$
Calculating this gives:
$MN = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25.$
Thus, the length of the line MN is 25.

To find the length of the line NP, we use the relation in triangle $NMP$ where the tangent at point $P$ creates a right triangle.
Using trigonometry and the cosine rule:
$\cos(NMP) = \frac{NP}{MN}$
From the triangle,
$NP = MN \cdot \cos(NMP)$
Finding the length gives us:
$NP = 25 \cdot \frac{7}{25} = 7.$
Thus, the length of the line NP is 24.