# Math Help - Two non-intersecting circles...

1. ## Two non-intersecting circles...

Two non-intersecting circles C1, containing points M and S, and C2, 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 is l1and the arc length NR is l2. The radius C1 is x, where x is greater or equal to 10 and the radius of C2 is 10.

(a) Explain why x>40.
---> Because if the radius of C2 is 10, then PQ minus this radius is 40. If the radius of C1 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 by l1+l2+MN+SR.
(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.