4.1 Circles The First Conic Section

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4.1 CirclesThe First Conic Section

• The basics, the definition, the formula
• And Tangent Lines

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The Standard Form of a circle comes from the
Distance Formula

• What is the length of the hypotenuse below?

y2
y1
x1
x2
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The Circle

• Definition The set of all points in a plane a
  given distance r radius from a given point
  (h,k) center.

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The Circle

• The Picture

(x,y)
r
(h,k)
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The Circle

• The Formula

(x,y)
r
(h,k)
What if the center is the origin? What happens
to the formula?
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Practice problems

• Find the center and radius of the following

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What did you see on that last problem?

• Completing the square is the norm, not the
  exception. Get ready for it and if you need to,
  go back in your notes and look it up (Section
  2.7). Most problems in these next 4 sections
  (circles, ellipses, hyperbolas and parabolas)
  will require completing the square whenever you
  are asked to graph.
• The clue is, if there are any linear terms (uh,
  what??) then be prepared to complete the square.

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How to graph?

• This is probably the easiest part of it all.
  Essentially, plot the center then
• r
• And then label those 4 points. If r is
  irrational, then estimate its value and plot in
  the general correct area (and label with radical
  notation)

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Practice Problem

• Graph
• (um what do I do if the 2nd degree terms have
  coefficients???)

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Notes

1. Plan to complete the square
2. If the radius is irrational, use it anyway and
   just estimate its location.
3. If the 2nd degree terms have coefficients, factor
   them out, complete the square, then divide by
   coefficients.

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Now we will learn to find the equation of their
tangent lines

• What do you know about a tangent line?
• So, if it perpendicular to the radius, what do
  you know about the slopes.

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What does a tangent line look like?
(x,y)
r
(h,k)
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Does the radius matter?

• In any circle, does the radius matter when it
  comes to the equation of the line tangent?
• 1. Find the equation of the line tangent to the
  circle at
  (2,2)
• No. It only gives you a location for the tangent
  point. It doesnt have any effect on the
  equation itself.

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So, what is the rule?

1. Find the standard form of the circle.
2. Identify the center.
3. Find the slope of the radius from the center to
   the given point.
4. Use the slope perpendicular (negative reciprocal)
   with the point ON THE CIRCLE to find the equation
   of the line tangent.

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More examples

1. Find the equation of the line tangent to the
   circle at
   (2,7).
2. Find the equation of the line tangent to the
   circle at the point in the
   4th quadrant where x 4.