1. Circles - Special Cases
Mathematics 4
August 16, 2011
Mathematics 4 () Circles - Special Cases August 16, 2011 1/8

2. Circle Equations
What 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. Circle Equations
What 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. Degenerate Sets
Degenerate Sets
A case where a mathematical object belonging to a certain class (circles)
belongs also to a different, simpler class (point).
Degenerate Circles
A 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. Null Sets
Null Set Circle Equations
A 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. Examples
Circle Equations
Find 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. Examples
Circle Equations
Find 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. Pair Seatwork 1
Part I.
Identify if the following equations are circles, points, or null sets. Show all
solutions.
1 2x2 + 2y 2 − 6x + 10y + 9 = 0
2 9x2 + 9y 2 + 6x − 24y + 17 = 0
3 6x2 + 6y 2 − 9x + 4y + 7 = 0
Part II. Previous Topics
Find 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