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Geometry by Jeremy J. Gray, Matthew F. Esplen, David A. Brannan

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Appendix 3: Solutions to the Problems

Chapter 1

Section 1.1

1.  Here we use the standard formula for the equation of a circle of given centre and radius given in Theorem 1.

(a)  This circle has equation

(x − 0)2 + (y − 0)2 = 12,

which can be rewritten in the form

x2 + y2 = 1.

(b)  This circle has equation

(x − 0)2 + (y − 0)2 = 42,

which can be rewritten in the form

x2 + y2 = 16.

(c)  This circle has equation

(x − 3)2 + (y − 4)2 = 22,

which can be rewritten in the form

x2 + y2 − 6x − 8y + 21 = 0.

(d)  This circle has equation

(x − 3)2 + (y − 4)2 = 32,

which can be rewritten in the form

x2 + y2 − 6x − 8y + 16 = 0.

2.  Since the origin lies on the circle, its coordinates (0, 0) must satisfy the equation of the circle. Thus

02 + 02 + f · 0 + g · ...

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