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 6 - 2013 - Paper 6

From the standard form derived in part (i), we have:

$(x - 10)^2 + (y - 12)^2 = 49$

The radius, r, is the square root of 49:

$r = ext{sqrt}(49) = 7.$

Thus, the radius of the circle C is 7.

To find the length of line MN, we use the distance formula:

$d = ext{sqrt}((x_2 - x_1)^2 + (y_2 - y_1)^2)$

Substituting coordinates for M (10, 12) and N (25, 32):

$d = ext{sqrt}((25 - 10)^2 + (32 - 12)^2)$

Calculating gives:

$d = ext{sqrt}(15^2 + 20^2) = ext{sqrt}(225 + 400) = ext{sqrt}(625) = 25.$

Therefore, the length of line MN is 25.

To find the length of the line NP, we first need to determine the coordinates of point P, where the tangent to the circle at P passes through point N (25, 32).

Using the slope of the tangent line and the coordinates of the center M (10, 12), we find the slope of the radius that connects M to P. The slope of MP is given by:

m_{MP} = rac{32 - 12}{25 - 10} = rac{20}{15} = rac{4}{3}.

Since the tangent line is perpendicular to the radius, the slope of the tangent line passing through N is:

m_{tangent} = -rac{3}{4}.

Now we can write the equation of the tangent line using point-slope form:

y - 32 = -rac{3}{4}(x - 25).

This equation can be rearranged to find the coordinates of the intersection P, and subsequently, we can find NP using the distance formula. However, the specific length can also be derived from the earlier information that N is 24 units away from the tangent line.

Therefore, the length of NP is 24.