TOPIC 4.2: ANALYTIC GEOMETRY

TOPIC 4.2: ANALYTIC GEOMETRY

4.2.1 The Ellipse

4.2.2 The Parabola

4.2.3 The Hyperbola

4.2.1 The Ellipse

aFPdPFd 2),(),( 21 =+

Definition of an ellipse:An ellipse is the set of all points in a plane the sum of whose

distances from 2 fixed points, F1 and F2 is constant. These 2 fixed points are called the foci.

The midpoint of the segment connecting the foci is the center of the ellipse

, a is a constant

Standard forms of the equations

of an ellipse

12

2

2

2

=+b

y

a

x )0,(),0,( cc− )0,(),0,( aa−

12

2

2

2

=+a

y

b

x),0(),,0( cc− ),0(),,0( aa−

The standard form of the equation of an ellipse with center at the origin, and major and minor axes of length 2a and 2b (Where a and b are positive and a2 > b2) is

Equation Foci Vertices Major axis

x-axis

y-axis

The foci are on the major axis, c units from the center, 222 bac −=

Graphing an ellipse centered at the origin

1936

22

=+ yx

144916 22 =+ yx

Example: 1-Graph and locate the foci:

2- Graph and locate the foci:

Finding the equation of an ellipse from its foci and vertices

Example:Find the standard form of the equation of an ellipse with foci at (-2, 0) and (2, 0) and vertices (-3, 0) and (3, 0)

Translation of ellipsesEllipses with Center at (h, k), a > b > 0,

and b2= a2 - c2

1)()(

2

2

2

2

=−+−b

ky

a

hx),( kch ± ),( kah ± ky =

1)()(

2

2

2

2

=−+−a

ky

b

hx ),( ckh ± ),( akh ± hx =

Equation Foci Vertices Major Axis

Graphing an ellipse centered at

(h, k)

Example: Graph

14

)2(

9

)1( 22

=−++ yx

Where are the foci located?

4.2.2 The Parabola

Definition of a parabola:A parabola is defined as the collection of all points P in the

plane that are the same distance from a fixed point F (focus) as they are from a fixed line D (directrix).

),(),( DPdPFd =

Axis of symmetry - the line through the focus F and perpendicular to the directrix D.

Vertex V - the point of intersection of the parabola with its axis of symmetry.

pxy 42 = pyx 42 =

Standard form of the equations of a parabolaThe standard form of the equation of a parabola with vertex at the origin is

or

The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. The length of lactus rectum is

|4p|

Finding the focus and directrix of a parabola

xy 82 =

yx 122 −=

Example:1.Find the focus and directrix of the parabola given by

2. Find the focus and directrix of the parabola given by

3. Find the standard form of the equation of a parabola with focus (8, 0) and directrix x = -8

. Then graph the parabola

. Then graph the parabola

Translation of parabola

Parabolas with Vertex at (h, k), p > 0

Equation Focus Directrix Axis if Symmetry

Description

Opens to right.

Opens to left.

Opens up.

Opens down.

)(4)( 2 hxpky −=− ),( khp + hpx +−= ky =

)(4)( 2 hxpky −−=− ),( khp +− hpx += ky =

)(4)( 2 kyphx −=− ),( kph + kpy +−= hx =

)(4)( 2 kyphx −−=− ),( kph +− kpy += hx =

Graphing parabola with vertex (h, k)

)1(4)2( 2 +=− yx

07422 =−++ xyy

Example:1.Find the vertex, focus, and directrix of the parabola given by

2. Find the vertex, focus and directrix of the parabola given by

. Then graph the parabola

4.2.3 The Hyperbola

aPFdPFd 2),(),( 21 ±=−

Definition of a hyperbola:A hyperbola is the collection of all points in the plane,

the difference of whose distances from two fixed points, called the foci, is a constant.

The midpoint of the segment connecting the foci is the center of the ellipse

, a is a constant

• Transverse Axis - the line containing the foci.

• Center - the midpoint of the line segment joining the foci.

• Conjugate Axis - the line through the center and perpendicular to the transverse axis.

• Branches – are symmetric with respect to the transverse axis, conjugate axis and center.

• Vertices - the two points of intersection of the hyperbola and the transverse axis.

Note:The hyperbola is symmetric with respect to its transverse axis and with respect to its conjugate axis.

Finding an equation of the hyperbola with center at the origin, one focus at (3, 0), and one vertex at (–2, 0). Graph the equation.

Distance from center to focus is c = 3Distance from center to vertex is a = 2.

( )( ) ( )

( )

Center: 0,0

Vertices: 4,0 , 4,0

Foci: 2 5,0

−

±

( )( ) ( )

( )

Center: 0,0

Vertices: 0, 2 , 0,2

Foci: 0, 5

−

±

Find an equation of the hyperbola having one vertex at (0, 2) and foci at (0, –3) and (0, 3). Graph the equation.

Looking at the points given we see that the center is at (0,0) and the transverse axis is along the y-axis.

Find an equation for the hyperbola with center at (1, –2), one focus at (4, –2), and one vertex at (3, –2). Graph the equation by hand.

Center, focus and vertex are on y = –2 so transverse axis is parallel to x-axis.

Distance from center to a focus is c = 3Distance from center to a vertex is a = 2