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Circles - Degenerate and Null cases

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Circles - Degenerate and Null cases

1. 1. Circles - Special Cases Mathematics 4 August 16, 2011Mathematics 4 () Circles - Special Cases August 16, 2011 1/8
2. 2. Circle EquationsWhat is strange about the following circle equations?Hint: Convert the equations into standard form. 1 x2 + y 2 − 8x + 6y + 25 = 0 2 x2 + y 2 + 10x − 14y + 78 = 0 Mathematics 4 () Circles - Special Cases August 16, 2011 2/8
3. 3. Circle EquationsWhat is strange about the following circle equations?Hint: Convert the equations into standard form. 1 x2 + y 2 − 8x + 6y + 25 = 0 → (x − 4)2 + (y + 3)2 = 0 2 x2 + y 2 + 10x − 14y + 78 = 0 → (x + 5)2 + (y − 7)2 = −4 Mathematics 4 () Circles - Special Cases August 16, 2011 3/8
4. 4. Degenerate SetsDegenerate SetsA case where a mathematical object belonging to a certain class (circles)belongs also to a diﬀerent, simpler class (point).Degenerate CirclesA point is a degenerate circle (circle equation with zero radius).e.g. (x − 4)2 + (y + 3)2 = 0 Mathematics 4 () Circles - Special Cases August 16, 2011 4/8
5. 5. Null SetsNull Set Circle EquationsA circle equation with an imaginary radius is a null set.e.g. (x + 5)2 + (y − 7)2 = −4 Mathematics 4 () Circles - Special Cases August 16, 2011 5/8
6. 6. ExamplesCircle EquationsFind the values of k so that the graph of x2 + y 2 + 8x − 12y = k + 3 is: 1 a circle 2 a point 3 a null set Mathematics 4 () Circles - Special Cases August 16, 2011 6/8
7. 7. ExamplesCircle EquationsFind the values of k so that the graph of x2 + y 2 + 8x − 12y = k + 3 is: 1 a circle → k > −55 2 a point → k = −55 3 a null set → k < −55 Mathematics 4 () Circles - Special Cases August 16, 2011 7/8
8. 8. Pair Seatwork 1Part I.Identify if the following equations are circles, points, or null sets. Show allsolutions. 1 2x2 + 2y 2 − 6x + 10y + 9 = 0 2 9x2 + 9y 2 + 6x − 24y + 17 = 0 3 6x2 + 6y 2 − 9x + 4y + 7 = 0Part II. Previous TopicsFind the equations of all circles having a radius whose endpoints are(−4, 5) and (2, −4). Show all solutions. Mathematics 4 () Circles - Special Cases August 16, 2011 8/8