10.2 The circle and the ellipse

M ATT KWA K

10.2 THE CIRCLE AND THE ELLIPSE

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 + (y-k)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