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} = 1^{2},

which can be rewritten in the form

*x*^{2} + *y*^{2} = 1.

(b) This circle has equation

(*x* − 0)^{2} + (*y* − 0)^{2} = 4^{2},

which can be rewritten in the form

*x*^{2} + *y*^{2} = 16.

(c) This circle has equation

(*x* − 3)^{2} + (*y* − 4)^{2} = 2^{2},

which can be rewritten in the form

*x*^{2} + *y*^{2} − 6*x* − 8*y* + 21 = 0.

(d) This circle has equation

(*x* − 3)^{2} + (*y* − 4)^{2} = 3^{2},

which can be rewritten in the form

*x*^{2} + *y*^{2} − 6*x* − 8*y* + 16 = 0.

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

0^{2} + 0^{2} + *f* · 0 + *g* · ...

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