Conics Instructional Unit

 

Day 6  - Circles

 

by

 

Mandy Stein

 

Circle

 

The locus of all points equal distance from a fixed point, called the center

 

Standard Equation of a Circle

With center (h , k)

(x – h) 2 + (y - k) 2 = r2

 

 

 

 

To graph a circle:

  1. Identify the center of the circle.
  2. Identify the radius
  3. Plot the center and a few points the distance of the radius from the center

 

(x – 2) 2 + y2 = 16

Center: (2,0)

Radius:  = 4

 

 

 

 

(x – 4)2 + (y + 3) 2 = 25

Center: (4,-3)

Radius:  = 5

 

 

 

 

To graph an equation not in standard form:

  1. Write the equation in standard form by completing the square
  2. Identify the center
  3. Identify the radius
  4. Plot the center and a few points the distance of the radius from the center

 

x2 + y2 + 4x – 6y – 3 = 0

 

First, we put the equation in standard form by completing the square

x2 + y2 + 4x – 6y = 3

(x2 + 4x + 4) + (y2 – 6y + 9) = 3 + 4 + 9

(x + 2) 2 + (y – 3) 2 = 16

Center: (-2,3)

Radius:  = 4

 

 

To write the equation of a circle:

  1. Identify the center of the circle
  2. Identify the radius
  3. Obtain the value of h, k, and r
  4. Substitute h, k, and r into the equation and simplify

 

Center: (4, 2)

Radius: 3

(x - 4) 2 + (y – 2) 2 = 9

 

 

Determine if the point (1,5) is inside, on, or outside the circle given by the equation (x – 2) 2 + (y +1) 2 = 9

 

To solve this problem, first determine the radius of the circle.  The radius is the square root of 9, which equals 3.  We also determine the center of the circle, which is (2, -1).  Then we determine how far (1,5) is from the center using the distance formula

d =

The distance between the center of the circle and the point (1,5) is .  Since this is larger than 3 (the radius of the circle) the point is outside the circle.

Day 7 - Introduction to Hyperbolas

 

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