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CONIC SECTIONS:

Y

PARABOLA

Line of symmetry: x=h 2p (h, k+p) p 2p

Focus

(h, k) p

. .

Vertex

Directrix

y=k-p X

Standard Equation Vertical Horizontal (x ­ h)2 = 4p(y ­ k) (y ­ k)2 = 4p(x ­ h)

Vertex (h, k) (h, k)

Focus (h, k + p) (h + p, k)

Directrix y=k­p x=h­p

Every point on a parabola is the same distance from the focus as from the directrix.

Y d1 2p p

.

P

d2 2p (h+p, k)

d1 = d2 Line of symmetry: y=k

(h, k)

. . .

p

Focus Vertex

Directrix

x=h-p X

Department of Mathematics, Sinclair Community College, Dayton, OH

1

CONIC SECTIONS:

P d1

ELLIPSE

.

c

(h, k+b)

d2 Center (h, k)

Vertex

(h-a, k)

Vertex

(h+c, k) a

.

(h-c, k)

.

.

.

.

(h+a, k)

Focus

Minor Axis

(Length = 2b)

b

Major Axis

(Length = 2a)

Focus

(h, k-b)

Standard Equation Horizontal Vertical Eccentricity: Center (h, k) (h, k) Foci (h+c, k), (h-c, k) (h, k+c), (h, k-c) c2 = a2 ­ b2

(graphs not to scale)

Vertices (h+a, k), (h-a, k) (h, k+a), (h, k-a)

(x - h) (y - k ) + =1 a2 b2 (x - h) 2 (y - k ) 2 + =1 b2 a2

2 2

c e= <1 a

a>b

For any point P on an ellipse, the sum of the distances from P to the foci is a constant: d1 + d2 = 2a. If a = b, then c = 0, and the ellipse is a circle.

(h, k+a)

Focus

. . . . .

Vertex

(h, k+c)

c Center (h, k) (h-b, k) b a (h+b, k)

(h, k-c)

Focus

Vertex

2

Department of Mathematics, Sinclair Community College, Dayton, OH

(h, k-a)

CONIC SECTIONS:

HYPERBOLA

Conjugate Axis x=h

P d1

Focus

·

Vertex

d2

a

(h-c, k)

.

(h-a, k)

.

c

(h, k) Center

. .

b

Vertex

(h+a, k)

(h+c, k)

.

Focus

Transverse Axis y=k

y = k + (b/a)(x ­ h)

y = k ­ (b/a)(x ­ h)

Standard Equation Transverse axis parallel to x-axis Transverse axis parallel to y-axis Eccentricity:

Center (h, k)

Foci (h+c, k), (h­c, k) (h, k+c), (h, k-c) c2 = a2 + b2

(graphs not to scale)

Vertices (h+a, k), (h-a, k)

Slant Asymptotes y=k± y=k±

(x - h) (y - k ) - =1 2 a b2

2 2

b (x-h) a a (x-h) b

(y - k ) 2 (x - h) 2 - =1 a2 b2

(h, k)

(h, k+a), (h, k-a)

c e= >1 a

For any point P on a hyperbola, the difference of the distances of P to the foci is a constant: d2 ­ d1 = 2a

Transverse Axis x=h

(h, k+c)

y = k ­ (a/b)(x-h)

c

(h, k+a)

(h, k)

y = k + (a/b)(x-h)

(h, k-a) (h, k-c) Focus

. . . . .

Focus

Vertex

a

Conjugate Axis y=k

b

Vertex

Department of Mathematics, Sinclair Community College, Dayton, OH

3

Another important type of hyperbola is defined by the equation xy = C, where C is a constant. The asymptotes are the x and y axes. The function is symmetric about the origin, and the transverse axis is y = x. (If C is negative, the transverse axis is y = -x, and the graph is in Quadrants II and IV.) This graph can also be viewed as a standard, x-y oriented hyperbola that has been rotated 45° into a new coordinate system.

. Focus: ( 2C , 2C ) . Vertex: ( C , C ) . .

____________________________________________________________________________

THE SECOND-DEGREE EQUATION

The ellipse/circle, parabola, and hyperbola functions are all cases of the second-degree equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Ax2 A Circle Ellipse Parabola Hyperbola Center/vertex not at origin Rotated axis 0 =C C = 0 if C 0 opp. sign of C + Bxy + B =0 =0 =0 =0 Cy2 C =A A = 0 if A 0 opp. sign of A 0 and/or 0 + Dx D + Ey E + F F

To rotate axes (eliminate the xy term): = angle of rotation from x-axis cot 2 = (A ­ C)/B x = x'cos ­ y'sin y = x'sin + y'cos Substitute A'(x')2 + C'(y')2 + D'x' + E'y' + F' = 0 Invariants: F' = F A + C = A' + C' B2 ­ 4AC = (B')2 ­ 4A'C' B2 ­ 4AC < 0 ellipse/circle B2 ­ 4AC = 0 parabola B2 ­ 4AC > 0 hyperbola

Department of Mathematics, Sinclair Community College, Dayton, OH

4

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