circles

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circles

  1. 1. A circle is the collection of points equidistant from a fixed point. The fixed point iscalled the center. The distance from the center to any point on the circle is theradius of the circle, and a segment containing the center whose end points are bothon the circle is a diameter of the circle. The radius, r, equals one-half the diameter,d.
  2. 2. The standard equation for a circle is (x - h)2 + (y - k)2 = r2. The center is at (h,k). The radius is r.In a way, a circle is a special case of an ellipse. Consider an ellipse whose fociare both located at its center. Then the center of the ellipse is the center ofthe circle, a = b = r, ande = = 0
  3. 3. In Euclidean geometry, a circle is the set of all points ina plane at a fixed distance, called the radius, from agiven point, the centre.Circles are simple closed curves which divide the planeinto an interior and exterior.The circumference of a circle is the perimeter of the circle, and the interior ofthe circle is called a disk. An arc is any continuous portion of a circle.A circle is a special ellipse in which the two foci coincide (i.e., are the samepoint). Circles are conic sections attained when a right circular cone isintersected with a plane perpendicular to the axis of the cone.
  4. 4. ContentsContents► Analytic resultsAnalytic results Equation of a circleEquation of a circle SlopeSlope Pi (π)Pi (π) CircumferenceCircumference Area enclosedArea enclosed
  5. 5. Analytic resultsAnalytic resultsEquation of a circleIn an x-y coordinate system, the circle with centre (a, b) and radius r is the set of allpoints (x, y) such that(x – a)2+ (y – b)2= r2The equation of the circle follows from the Pythagorean theorem applied to any pointon the circle.If the circle is centred at the origin (0, 0), then this formula can be simplified toX2+ y2= r2and its tangent will bexx1 + yy1 = r2where x1, y1 are the coordinates of the common point.When expressed in parametric equations, (x, y) can be written using thetrigonometric functions sine and cosine asx = a + r cost,y = b + r sintwhere t is a parametric variable, understood as the angle the ray to (x, y) makes withthe x-axis.
  6. 6. In homogeneous coordinates each conic section with equation of a circle isax2+ ay2+ 2b1xz + 2b2yz + cz2= 0It can be proven that a conic section is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the circular points at infinity.In polar coordinates the equation of a circle isr2– 2rr0 cos(θ - φ ) + r02 = a2
  7. 7. SlopeThe slope of a circle at a point (x, y) can be expressed with the following formula,assuming the centre is at the origin and (x, y) is on the circle:y´ = - x/yMore generally, the slope at a point (x, y) on the circle(x − a)2+ (y − b)2= r2i.e., the circle centred at (a, b) with radius r units, is given byy´ = (a – x) / (y – b)provided that y ≠ b, of course.
  8. 8. Pi (π)Pi or π is the ratio of a circles Circumference to its Diameter.π = C/D ≈ 3.141592654The numeric value of π never changes.π is always approximately 3.14159.In modern English, it is pronounced /paɪ/ (as in apple pie).
  9. 9. CircumferenceThe distance around a circle is called its circumference. The distance across acircle through its center is called its diameter. We use the Greek letter ∏(pronounced Pi) to represent the ratio of the circumference of a circle to thediameter. In the last lesson, we learned that the formula for circumference of acircle is: . For simplicity, we use = 3.14. We know from the last lesson that thediameter of a circle is twice as long as the radius. This relationship is expressedin the following formula: .d = 2 · r
  10. 10. AreaThe area of a circle is the number of square units inside that circle. If eachsquare in the circle to the left has an area of 1 cm2, you could count the totalnumber of squares to get the area of this circle. Thus, if there were a total of28.26 squares, the area of this circle would be 28.26 cm2 However, it is easierto use one of the following formulas:Where A is the area, and r is the radius. Lets look at some examplesinvolving the area of a circle. In each of the three examples below, we willuse ∏ = 3.14 in our calculations.

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