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MATT KWAK 10.2 THE CIRCLE AND THE ELLIPSE
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CIRCLE Set of all points in a plane that are at a fixed distance from a fixed point(center) in the plane. With the center (a,b) and radius r, standard equation of a circle is (x-a) 2 + (y-b) 2 = r 2
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EXAMPLE Find the center and the radius of an equation and graph it x 2 + y 2 -16x + 14y + 32 = 0
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x 2 -16x + y 2 + 14y = -32 x 2 -16x +64 + y 2 + 14y + 49 = -32 +64 +49 (x-8) 2 + (y+7) 2 = 9 2 So the center is (8,-7) and the radius is 9. But to graph it we need to make it something looks like y= ~~~
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(x-8) 2 + (y+7) 2 = 81 (y+7) 2 = 81- (x-8) 2 y+7 = ±√(81- (x-8) 2 ) y = -7 ±√(81- (x-8) 2 )
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ELLIPSE It is the set of all points in a plane. The Sum of whose distances from two fixed points( the foci) is constant. The center is the midpoint of the segment between the foci.
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Major Axis Horizontal graph and Standard Equation Major Axis Vertical graph and Standard Equation C 2 = a 2 – b 2
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EXAMPLE Find the standard equation of the ellipse with vertices (-5, 0) and (5,0) and foci (-3,0) and (3,0) then graph it.
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Standard Equation: x 2 /a 2 + y 2 /b 2 = 1 C 2 = a 2 – b 2 3 2 = 5 2 – b 2 b 2 = 16 Standard Equation: x 2 /25 + y 2 /16 = 1 y= ±√(400 – 16x 2 /25)
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ELLIPSE WITH THE CENTER Axis Horizontal: (x-h) 2 /a 2 + (y-k) 2 /b 2 = 1 Axis Vertical: (x-h) 2 /b 2 + (y- k) 2 /a 2 = 1
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EXAMPLE For the ellipse equation 4x 2 + y 2 + 24x -2y + 21 =0, find the center and then graph it.
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4x 2 + y 2 + 24x -2y + 21 =0 4x 2 + 24x + y 2 -2y =-21 4(x 2 + 6x + 9)+ (y 2 -2y + 1) =-21 +4 × 9 +1 4(x +3) 2 + (y-1) 2 = 16 1/16 × [4(x +3) 2 + (y-1) 2 ] = 16 × 1/16 (x + 3) 2 /4 + (y-1) 2 /16 = 1 [x-(-3)] 2 /2 2 + (y-1) 2 /4 2 = 1 Center : ( -3, 1) y= 1± 2√4-(x+3) 2
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