Distance and Midpoint Formulas; Circles

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Distance and Midpoint Formulas Circles
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The Distance Formula

• The distance, d, between the points (x1, y1) and
  (x2,y2) in the rectangular coordinate system is

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Example

• Find the distance between (-1, 2) and (4, -3).

Solution Letting (x1, y1) (-1, 2) and (x2,
y2) (4, -3), we obtain
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The Midpoint Formula

• Consider a line segment whose endpoints are (x1,
  y1) and (x2, y2). The coordinates of the
  segment's midpoint are
• To find the midpoint, take the average of the two
  x-coordinates and of the two y-coordinates.

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Text Example

• Find the midpoint of the line segment with
  endpoints (1, -6) and (-8, -4).
• Solution To find the coordinates of the midpoint,
  we average the coordinates of the endpoints.
• (-7/2, -5) is midway between the points (1, -6)
  and (-8, -4).

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Definition of a Circle
A circle is the set of all points in a plane that
are equidistant from a fixed point called the
center. The fixed distance from the circles
center to any point on the circle is called the
radius.
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The Standard Form of the Equation of a Circle
The standard form of the equation of a circle
with center (h, k) and radius r is (x h)2
(y k)2 r2.
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Example
Find the center and radius of the circle whose
equation is (x 2)2 (y 3)2 9 and graph
the equation.
Solution In order to graph the circle, we
need to know its center, (h, k), and its radius
r. We can find the values of h, k, and r by
comparing the given equation to the standard form
of the equation of a circle.
(x 2)2 (y 3)2 9
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Example cont.
Find the center and radius of the circle whose
equation is (x 2)2 (y 3)2 9 and graph
the equation.
Solution
We see that h 2, k -3, and r 3. Thus, the
circle has center (2, -3) and a radius of 3
units. Plot the center, (2, -3), and find 3
additional points by going up, right, down, and
left of the center by 3 units.
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General Form of the Equation of a Circle

• The general form of the equation of a circle is
• x2 y2 Dx Ey F 0.
• Complete the square

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Example-Completing the Square

• Given the equation