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

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

The equation of a circle in standard form is (x – h) 2 + (y – k) 2 = r 2

Example 1 Find the center and radius of each circle a) ( x – 11 )² + ( y – 8 )² = 25 b) ( x – 3 )² + ( y + 1 )² = 81 c) ( x + 6 )² + y ² = 21 Center = ( 11,8 ) Radius = 5 Center = ( 3,-1 ) Radius = 9 Center = ( -6,0 ) Radius = 21

Example 2 Find the equation of the circle in standard form:

Example 3 Find the equation of the circle with centre (–3, 4) and passing through the origin.

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

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

Example 4: Write the equation in standard form and find the center and radius length of : Group terms Complete the square a)

b)

Inequalities of a Circle Example 5: Determine the inequality that represents the shaded region

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 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

 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

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).

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(-½, ½). 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/ (x + 1) 2 + (y + 2) 2 = 13/2 Center (-1,-2) and P(-½, ½)