Two non-intersecting circles C_{1}, containing points M and S, and C_{2}, containing points N and R, have centres P and Q where PQ=50. The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is α, the size of the obtuse angle NQR is β, and the size of the angle MPQ is θ. The arc length MS isl_{1}and the arc length NR isl. The radius C_{2}_{1 }is x, where x is greater or equal to 10 and the radius of C_{2}is 10.

(a) Explain why x>40.

---> Because if the radius of C_{2}is 10, then PQ minus this radius is 40. If the radius of C_{1}exceeds 40, then PQ would not make sense and it would have to have another value.

(b) Show that cosθ = (x-10)/50.

(c) (i) Find an expression for MN in terms of x.

(i) Find the value of x that minimises MN.

(d) Find an expression in terms of x for:

(i) α;

(ii) β.

(e) The length of the perimeter is given byl+MN+SR._{1}+l_{2}

(i) Find an expression, b(x), for the length of the perimeter in terms of x.

(ii) Find the maximum value of the length of the perimeter.

(iii) Find the value of x that fives a perimeter of length 200.

Please help; any contribution would be a lot of help