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Two circles, A and B, touch each other at exactly one point, as shown in the diagram below. The equation of circle A is . Circle B has centre (1, k + 1) and radius 4.

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(Not to scale)

a)
i) Find the completed square form of x2 - 18x
Find the completed square form of y2 - 2y

ii) Use your answer from part i) to express the equation of circle A in the form
(x - a)2 + (y - b)2 = r2

iii) write down the centre and radius of circle A.

b)
i) Explain why the distance between the centres of circles A and B is 10.

ii) Express this distance also in terms of k, and hence find the two possible values of k.

iii) Hence show that the two possible equations of circle B are

and

c) Show algebraically that the line y = 4/3 x - 1 is a tangent to circle A, and give the coordinates of the point of contact.

(You should work with fractions in this part of the question)

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