Circles - Special Cases

Mathematics 4

August 16, 2011

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Circle Equations

What is strange about the following circle equations?

Hint: Convert the equations into standard form.

1 x2 + y2 − 8x+ 6y + 25 = 0

2 x2 + y2 + 10x− 14y + 78 = 0

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Circle Equations

What is strange about the following circle equations?

Hint: Convert the equations into standard form.

1 x2 + y2 − 8x+ 6y + 25 = 0→ (x− 4)2 + (y + 3)2 = 0

2 x2 + y2 + 10x− 14y + 78 = 0→ (x+ 5)2 + (y − 7)2 = −4

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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

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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

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Examples

Circle Equations

Find the values of k so that the graph of x2 + y2 + 8x− 12y = k + 3 is:

1 a circle

2 a point

3 a null set

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Examples

Circle Equations

Find the values of k so that the graph of x2 + y2 + 8x− 12y = k + 3 is:

1 a circle → k > −552 a point → k = −553 a null set → k < −55

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Pair Seatwork 1

Part I.

Identify if the following equations are circles, points, or null sets. Show allsolutions.

1 2x2 + 2y2 − 6x+ 10y + 9 = 0

2 9x2 + 9y2 + 6x− 24y + 17 = 0

3 6x2 + 6y2 − 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.

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