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Intro to Conics - Circles

Intro to Conics - Circles. I. . Parts of a Circle. A) Standard form of a circle equation: x 2 + y 2 = r 2 1) In a later section we will write the equation like this… (x – h) 2 + (y – k) 2 = r 2 B) The center is at ( 0 , 0 ).

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Intro to Conics - Circles

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  1. Intro to Conics - Circles I.. Parts of a Circle. A) Standard form of a circle equation: x2 + y2 = r2 1) In a later section we will write the equation like this… (x – h)2 + (y – k)2 = r2 B) The center is at ( 0 , 0 ). 1) If in (x – h)2 + (y – k)2 = r2 form, the center is (h , k). 2) Set the inside = 0 and solve (change the sign). C) The radius of the circle can be found by taking the square root of the # on the end ( = r ).

  2. Intro to Conics - Circles II.. Writing Equations of Circles in Standard Form x2 + y2 = #. A) Get the x & y values on one side of the equal sign. 1) Move the # part to the other side of the equal sign. B) Divide both sides by the value in front of the x2 (if needed). Examples: Write the equation for a circle in standard form. 1) 6x2 – 30 = – 6y2 2) 8y2 + 38 = – 8x2 + 62 6x2 + 6y2 = 30 8x2 + 8y2 = 24 (divide by 6) (divide by 8) x2 + y2 = 5 x2 + y2 = 3

  3. Intro to Conics - Circles III.. Writing Equations of Circles from Points on the Circle. A) If you know the center (0 , 0) and any point on the graph, you can find the radius by using the distance formula. 1) Distance = 2) If the center is (0 , 0), then you have (x – 0)2 = x2 and (y – 0)2 = y2. So the distance = 3) But the distance is the radius. The circle equation says x2 + y2 = r2, so we have to square the distance. a) r2 = so r2 = (x)2 + (y)2. B) Now write the equation: x2 + y2 = #

  4. Intro to Conics - Circles IV.. Writing Equations of Circles given the Center & the Radius. A) If the center is at the origin (0 , 0), then the standard equation is x2 + y2 = #. 1) If the center is NOT at the origin, then you change the signs to put it in (x – h)2 + (y – k)2 = # form. Example: Center is (5 , – 8), then eq: (x – 5)2 + (y + 8)2 = # B) If the center is at the origin (0 ,0) and you are told the value of r, then square that value to get r2 (write it twice). Example: r = 8, so 82 = 64 r = , so x2 + y2 = 64 x2 + y2 = 45

  5. Intro to Conics - Circles V.. Graphing a Circle that is written in Standard Form. A) Find and graph the center of the circle ( 0 , 0 ). 1) If in (x – h)2 + (y – k)2 = # form, the center is (h , k). B) Find the radius of the circle. 1) Square root the end number. C) Graph the “vertices” of the circle. 1) From the center, graph a dot that has a length of “r” up, down, left and right of the center point. 2) Connect these four points with a round line. HW: Circles # 3 – 8 all, 9 – 19 odd, 23 – 29 odd, 45 – 51 odd

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