f4 maths circle

用 intersecting secants theorem.
(http://www.mathopenref.com/secantsintersecting.html)

a.
let OC cut the circle at E, then
OE.EC=AB.BC
5*EC = 8*10
EC=8*10/5=16
distance from point B to centre of circle O
= OC
= OE+EC
=5+16
=21

b.
Longest distance occurs when points A and B merge together at the point T, or
when the line CT is the tangent to the circle.
Thus
at the tangent, using Pythagoras' theorem,
CT^2=OC^2-R^2
=21^2-5^2
=441-25
=416
Length of tangent
=CT
=sqrt(416)
=20.396...