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CIRCLES

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.

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CIRCLES

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  1. CIRCLES

  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

  3. Equation of a Circle in Standard Form 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

  4. The equation of a circle in standard form is (x – h)2 + (y – k)2 = r2

  5. 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

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

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

  8. Equation of a Circle in General Form The equation of a circle in general form is x2+ y2 + ax + by + c = 0 Only if a2 + b2 > 4c

  9. From General to Standard… • Group x terms together, y-terms together, and move constants to the other side • Complete the square for the x-terms • Complete the square for the y-terms • Remember that whatever you do to one side, you must also do to the other

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

  11. b)

  12. Inequalities of a Circle Example 5:Determine the inequalitythatrepresents the shadedregion

  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. Tangent and Secant Lines

  14. Tangent Line to a Circle l • Line l is tangent tothe circleat point P • P is the point of tangency x y r h k

  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 slopeperpendicular to the slope of the radius (negativereciprocal) • Use the point of tangency and negativereciprocal to determine the equation of the tangent line

  16. Example 6 Determine the equation of the tangent line to the circlewithequation (x-2)2 + (y-1)2 = 5 at the point (1,3).

  17. Example 7 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)

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