The circumferences of circles ­C₁ and C₂ touch at exactly one point.

Both circles also touch a straight line L, at points Q₁ and Q₂ respectively.

The centre of C₁ is P₁, and the centre of C₂ is P₂.

The radius of C₁ is R and the radius of C₂ is r.

(i) Find the area of the trapezium T with vertices Q₁P₁P₂Q₂ in terms of R and r

(ii) Given that R + r = 10 units, find the maximum area of T. Prove that your answer really is the maximum value.