Pc Conic Section

Conic Sections

Conic sections are plane figures formed by the intersection of a double-napped cone and a plane.

Parabola

Ellipse

Hyperbola

The conic sections may be defined as sets of points in the plane that satisfy certain geometrical properties.

A parabola is the set of all points in the plane equidistant from a fixed line and a fixed point not on the line. The fixed line is the directrix. The fixed point is the focus. The axis is the line passing through the focus and perpendicular to the directrix. The vertex is the midpoint of the line segment along the axis joining the directrix to the focus.

axis parabola focus

directrix vertex

The standard form for the equation of a parabola with vertex at the origin and a vertical axis is: x2 = 4py where p ≠ 0 vertical axis: x = 0

directrix: y = –p,

focus: (0, p)

y (0, p) p

x2 = 4py x

(0, 0) y = –p

Note: p is the directed distance from the vertex to the focus.

The standard form for the equation of a parabola with vertex at the origin and a horizontal axis is: y2 = 4px where p ≠ 0 horizontal axis: y = 0, directrix: x = –p focus: ( p, 0) y y2 = 4px (0, 0) x p (p, 0) x = –p

Note: p is the directed distance from the vertex to the focus.

Example: Find the directrix, focus, and vertex, and sketch the 1 2 y = − x . parabola with equation 8 Rewrite the equation in standard form x2 = 4py. x2 = – 8y → x2 = 4(–2)y → p = –2 y vertex: (0, 0) vertical axis: x = 0 directrix: y = – p → y = 2

y=2 (0, 0) (0, –2)

focus: = (0, p) → (0, –2) x=0

x 1 2 y=− x 8

Example: Write the standard form of the equation of the parabola with focus (1, 0) and directrix x = –1. y

p=1 (1, 0) (0, 0) vertex

x

x = -1 Use the standard from for the equation of a parabola with a horizontal axis: y2 = 4px. p = 1 → y2 = 4(1)x. The equation is y2 = 4x.

An ellipse is the set of all points in the plane for which the sum of the distances to two fixed points (called foci) is a positive constant. The major axis is the line segment passing through the foci with endpoints (called vertices) on the ellipse. The midpoint of the major axis is the center of the ellipse.

vertex The minor axis is the line segment perpendicular to the major axis passing through the center of the ellipse with endpoints on the ellipse.

Ellipse

focus

d1

center

d1 + d2 = constant d2 focus major axis vertex

minor axis

The standard form for the equation of an ellipse with center at x2 y2 the origin and a major axis that is horizontal is: 2 + 2 = 1 , with: a b vertices: (–a, 0), (a, 0) and foci: (–c, 0), (c, 0) where c2 = a2 – b2 y a c (– a, 0)

(–c, 0) (0, – b)

(0, b) b

x2 y2 + 2 =1 2 a b

x a (0, 0) (c, 0) (a, 0)

The standard form for the equation of an ellipse with center at x2 y2 the origin and a major axis that is vertical is: 2 + 2 = 1, with: b a vertices: (0, –a), (0, a) and foci: (0, –c), (0, c) where c2 = a2 – b2 y

a (– b, 0)

b

c

(0, a) x2 y2 (0, c) b 2 + a 2 = 1

(0, 0) x b (b, 0) a (0, -c) (0, – a)

Example: Sketch the ellipse with equation 25x2 + 16y2 = 400 and find the vertices and foci. y 1. Put the equation into standard form. 25 x 2 + 16 y 2 = 400 divide by 400 (0, 5) (0, 3) 2 2 25 2 16 2 x y x + y =1→ + =1 5 (4, 0) 3 400 400 16 25 x So, a = 5 and b = 4. (–4, 0) 4 2. Since the denominator of the y2-term (0, –3) (0, –5) is larger, the major axis is vertical. 3. Vertices: (0, –5), (0, 5) 4. The minor axis is horizontal and intersects the ellipse at (–4, 0) and (4, 0). 5. Foci: c2 = a2 – b2 → (5)2 – (4)2 = 9 → c = 3

foci: (0, –3), (0,3)

A hyperbola is the set of all points in the plane for which the difference from two fixed points (the foci) is a positive constant. The graph of the hyperbola has two branches.

hyperbola transverse axis

The line through the foci intersects the hyperbola at two points called vertices.

focus The line segment joining the vertices is the transverse axis. Its midpoint is the center of the hyperbola.

vertex d1

vertex focus d2

center d2 – d1 = constant

The standard form for the equation of a hyperbola with a 2 2 x y horizontal transverse axis is: 2 − 2 = 1 with: a b vertices: (– a, 0), (a, 0) and foci: (– c, 0), (c, 0) where b2 = c2 – a2 vertex (– a, 0)

asymptote

y (0, b)

y=

b x a

vertex (a, 0) focus x (c, 0) asymptote

focus (–c, 0) (0, –b)

b y=− x a

A hyperbola with a horizontal transverse axis has asymptotes b b y = x y = − x. with equations and a

a

The standard form for the equation of a hyperbola with a y2 x2 vertical transverse axis is: 2 − 2 = 1 with: a b vertices: (0, – a), (0, a) and foci: (0, – c), (0, c) where b2 = c2 – a2 y asymptote a focus (0, c) y= x vertex b (0, a) (–b, 0) vertex (0, – a)

x (b, 0) asymptote

focus (0, -c)

a y=− x b

A hyperbola with a vertical transverse axis has asymptotes a a with equations y = x and y = − x . b b

Example: Sketch the hyperbola with equation x2 – 9y2 = 9 and find the vertices, foci, and asymptotes. 1. To write the equation in standard form, divide by 9. x2 y2 → 2 − 2 = 1 → a = 3, b = 1. 3 1 2. Because the x2-term is positive, the transverse axis is horizontal. 3. Vertices: (0, –3), (0, 3) 1 y = ± x 4. Asymptotes: 3 5. b 2 = c 2 − a 2

(−

10 , 0

)

foci: (− 10 , 0), ( 10 , 0)

( 10 , 0) 1 (0, 1) y= x 3

x

(1) 2 = c 2 − (3) 2 → c = ± 10

a = 3 and b = 1

y

(-3, 0)

1 y=− x 3 (0, -1) (3, 0)