Machinery's Handbook, 31st Edition
52 ELLIPSE FORMULAS Example 1: Determine the values of h , k , a , b , c , and e of the following ellipse: 3 x 2 5 y 2 12 x – 30 y 42 + + + = 0 Solution: Rearrange the ellipse equation into the standard form as follows, using the method of completing the square (see page 34): 3 x 2 5 y 2 12 x – 30 y 42 + + + 3 x 2 12 x – 5 y 2 30 y 42 + + + 0 = = 3 x 2 4 x – 2 2 + ( ) 5 y 2 6 y 3 2 + + ( ) + = 15 3 x – 2 ( ) 2 15 ----------- 5 y +3 ( ) 2 15 + ----------- x – 2 ( ) 2 5 ( ) 2 ---------- y +3 ( ) 2 3 ( ) 2 + ---------- 1 = = Comparing to the form x h – ( ) 2 a 2 ---------- y k – ( ) 2 b 2 + --------- = 1 and solving for c and e gives: h = 2 k = –3 a = 5 b = 3 c = 2 e , , Additional Formulas: An ellipse can be represented “parametrically” by the equations x = a cos q and y = b sin q , where x and y are the rectangular coordinates of any point on the ellipse, and the parameter q is the angle at the center measured from the x -axis counter- clockwise. The following figures correspond to the formulas below them for other mea - surements of the ellipse. , , , 2 5 = --
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