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Slide 1
Slide 2
Definitions Circle: The set of all points that are the same
distance from the center Radius: a segment whose endpoints are the
center and a point on the circle A circle is a group of points,
equidistant from the center, at the distance r, called a
radius
Slide 3
The center of a circle is given by (h, k) The radius of a
circle is given by r The equation of a circle with its centre at
the origin in standard form is x 2 + y 2 = r 2 Equation of a Circle
in Standard Form
Slide 4
The equation of a circle in standard form is (x h) 2 + (y k) 2
= r 2
Slide 5
Example 1 Find the center and radius of each circle a) ( x 11 )
+ ( y 8 ) = 25 b) ( x 3 ) + ( y + 1 ) = 81 c) ( x + 6 ) + y = 21
Center = ( 11,8 ) Radius = 5 Center = ( 3,-1 ) Radius = 9 Center =
( -6,0 ) Radius = 21
Slide 6
Example 2 Find the equation of the circle in standard
form:
Slide 7
Example 3 Find the equation of the circle with centre (3, 4)
and passing through the origin.
Slide 8
Equation of a Circle in General Form The equation of a circle
in general form is x 2 + y 2 + ax + by + c = 0 Only if a 2 + b 2
> 4c
Slide 9
From General to Standard 1. Group x terms together, y-terms
together, and move constants to the other side 2. Complete the
square for the x-terms 3. Complete the square for the y-terms
Remember that whatever you do to one side, you must also do to the
other
Slide 10
Example 4: Write the equation in standard form and find the
center and radius length of : Group terms Complete the square
a)
Slide 11
b)
Slide 12
Inequalities of a Circle Example 5: Determine the inequality
that represents the shaded region
Slide 13
Tangents and secants are LINES A tangent line intersects the
circle at exactly ONE point. A secant line intersects the circle at
exactly TWO points.
Slide 14
Tangent Line to a Circle Line l is tangent to the circle at
point P P is the point of tangency x y h k r l
Slide 15
To find the tangent line: Calculate the slope of the radius
line connecting the center to the point on the circle The tangent
line has a slope perpendicular to the slope of the radius (negative
reciprocal) Use the point of tangency and negative reciprocal to
determine the equation of the tangent line
Slide 16
Example 6 Determine the equation of the tangent line to the
circle with equation (x-2) 2 + (y-1) 2 = 5 at the point (1,3).
Slide 17
Example 7 Determine the equation of the tangent line to the
circle with equation 2x 2 + 2y 2 + 4x + 8y - 3 = 0 at the point
P(-, ). 2x 2 + 2y 2 + 4x + 8y 3 = 0 (2x 2 + 4x) + (2y 2 + 8y) = 3
(x 2 + 2x) + (y 2 + 4y) = 3/2 (x 2 + 2x + 1) + (y 2 + 4y + 4) = 3/2
+ 1 + 4 (x + 1) 2 + (y + 2) 2 = 13/2 Center (-1,-2) and P(-, )