10.2 THE CIRCLE AND
THE ELLIPSE
MATT KWAK
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
EXAMPLE
• Find the center and the radius of an equation and
graph it
x2 + y2 -16x + 14y + 32 = 0
x2 + y2 -16x + 14y + 32 = 0
x2 -16x
+ y2 + 14y
= -32
x2 -16x +64 + y2 + 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= ~~~
(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 )
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.
Major Axis Horizontal graph and
Standard Equation
Major Axis Vertical graph and
Standard Equation
C 2 = a 2– b 2
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.
Standard Equation: x2/a2 + y2/b2 = 1
C 2 = a 2– b 2
3 2 = 5 2– b 2
b 2 = 16
Standard Equation: x2/25 + y2/16 = 1
y= ±√(400 – 16x2/25)
ELLIPSE WITH THE CENTER
Axis Horizontal: (x-h)2/a2 +
(y-k)2/b2 = 1
Axis Vertical: (x-h)2/b2 + (yk)2/a2 = 1
EXAMPLE
• For the ellipse equation 4x2 + y2 + 24x -2y + 21 =0,
find the center and then graph it.
4x2 + y2 + 24x -2y + 21 =0
4x2 + 24x +
y2 -2y =-21
4(x2 + 6x + 9)+ (y2 -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)]2/22 + (y-1)2/42 = 1
Center : ( -3, 1)
y= 1± 2√4-(x+3)2