CIRCLES. 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.
A circle is a group of points, equidistant from the center, at the distance r, called a radius
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
x2+ y2= r2
The equation of a circle in standard form is
(x – h)2 + (y – k)2 = r2
Example 1Find the center and radius of each circle
a) ( x – 11 )² + ( y – 8 )² = 25
Center = ( 11,8 ) Radius = 5
b) ( x – 3 )² + ( y + 1 )² = 81
Center = ( 3,-1 ) Radius = 9
c) ( x + 6 )² + y ² = 21
Center = ( -6,0 ) Radius = 21
Find the equation of the circle in standard form:
Example 3Find the equation of the circle with centre (–3, 4) and passing through the origin.
The equation of a circle in general form is
x2+ y2 + ax + by + c = 0
Only if a2 + b2 > 4c
Complete the square
Example 5:Determine the inequalitythatrepresents the shadedregion
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.
Tangent and Secant Lines
Determine the equation of the tangent line to the circlewithequation (x-2)2 + (y-1)2 = 5 at the point (1,3).
Determine the equation of the tangent line to the circlewithequation2x2+ 2y2 + 4x + 8y - 3 = 0at the point P(-½, ½).
2x2 + 2y2 + 4x + 8y – 3 = 0 (2x2 + 4x) + (2y2 + 8y) = 3
(x2 + 2x) + (y2 + 4y) = 3/2
(x2 + 2x + 1) + (y2 + 4y + 4) = 3/2 + 1 + 4
(x + 1)2 + (y + 2)2 = 13/2
Center (-1,-2) and P(-½, ½)