By Conte A. (ed.)
Lecture notes in arithmetic No.947
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Extra resources for Algebraic Threefolds, Varenna, Italy 1981, Second Session: Proceedings
We will use ðÀ5ó 6Þ to ﬁnd r2 . ðÀ5 À 5Þ2 þ 62 ¼ r2 100 þ 36 ¼ r2 136 ¼ r2 The equation is ðx À 5Þ þ y2 ¼ 136. Equations of circles are not always written in the form ðx À hÞ2 þ ðy À kÞ2 ¼ r2 . For example, the equation ðx À 2Þ2 þ ðy þ 3Þ2 ¼ 16 might be written in its expanded form. ðx À 2Þ2 þ ðy þ 3Þ2 ¼ 16 ðx À 2Þðx À 2Þ þ ðy þ 3Þðy þ 3Þ ¼ 16 x2 À 4x þ 4 þ y2 þ 6y þ 9 ¼ 16 After using the FOIL method x2 þ y2 À 4x þ 6y À 3 ¼ 0 In the following, we will be given equations like the one above and use completing the square to rewrite them in the form ðx À hÞ2 þ ðy À kÞ2 ¼ r2 .
As before, we will isolate the absolute value expression on one side of the inequality, then solve the inequality. EXAMPLES * 3jx þ 4j À 7 ! 5 3jx þ 4j ! 12 jx þ 4j ! 4 x À8 1 8 À 4 x À 1 ! 0 2 ðÀ1ó À 8 [ ½0ó 1Þ 1 À4 x À 1 ! À5 2 À4ð1=2Þx À 1 À5 À4 À4 1 5 x À 1 2 4 CHAPTER 2 26 5 4 1 À 4 1 2 À 4 À À 1 2 1 xÀ1 2 1 9 x 2 4 1 2 x 2 x 9 2 Absolute Value 5 4 9 2 4 ! 1 9 À ó 2 2 PRACTICE Solve the inequality and give the solution in interval notation. 1.
X2 À 12x þ y2 þ 4y ¼ À36 53 CHAPTER 3 The xy Coordinate Plane 54 Next, we will complete the square for the x-terms and the y-terms and will add both numbers to each side of the equation. x2 À 12x þ 36 þ y2 þ 4y þ 4 ¼ À36 þ 36 þ 4 In the last step, we will write the left side of the equation as the sum 2 2 of þ 36 ¼ ðx À pﬃﬃﬃ Þ , we will use pﬃﬃﬃﬃﬃtwo perfect 2squares. For x À 12x 2 36 ¼ 6. For y þ 4y þ 4 ¼ ðy þ Þ , we will use 4 ¼ 2. ðx À 6Þ2 þ ðy þ 2Þ2 ¼ 4 * Now we can see that this equation is an equation of a circle which has center ð6ó À2Þ and radius 2.