1515 conics
TRANSCRIPT
Conics
Conic Sections(1) Circle
A circle is formed when
i.e. when the plane is perpendicular to the axis of the cones.
2
Conic Sections(2) Ellipse
An ellipse is formed when
i.e. when the plane cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.
2
Conic Sections(3) Parabola
A parabola is formed when
i.e. when the plane is parallel to a generator.
Conic Sections(4) Hyperbola
A hyperbola is formed when
i.e. when the plane cuts both the cones, but does not pass through the common vertex.
0
ParabolaA parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a).
focus F(a,0)
P(x,y)
M(-a,0) x
y
O
Form the definition of parabola,PF = PN
axyax 22)(222 )()( axyax
22222 22 aaxxyaaxx
axy 42 standard equation of a parabola
mid-point of FM = the origin (O) = vertex
length of the latus rectum = LL’= 4a
vertex
latus rectum (LL’)
axis of symmetry
Other forms of Parabola
axy 42
Other forms of Parabola
ayx 42
Other forms of Parabola
ayx 42
Ellipses
An ellipse is the locus of a variable point on a plane so that the sum of its distance from two fixed points is a constant.
P’(x,y)
P’’(x,y)
Let PF1+PF2 = 2a where a > 0
aycxycx 2)()( 2222 2222 )(2)( ycxaycx
222222 )()(44)( ycxycxaaycx
222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa
42222222222 22 acxaxcyacaxcaxa
22422222 )( caayaxca
)()( 22222222 caayaxca 222 cabLet
222222 bayaxb
12
2
2
2
by
ax standard equation of
an ellipse
vertex
major axis = 2a
minor axis = 2b
lactus rectum
length of semi-major axis = a
length of the semi-minor axis = b
length of lactus rectum = ab22
Other form of Ellipse
12
2
2
2
ay
bx
Hyperbolas
A hyperbola is the locus of a variable point on a plane such that the difference of its distance from two fixed points is a constant.
P’(x,y)
Let |PF1-PF2| = 2a where a > 0
aycxycx 2|)()(| 2222 2222 )(2)( ycxaycx
222222 )()(44)( ycxycxaaycx
222 44)(4 acxycxa 42222222 2)2( acxaxcycxcxa
42222222222 22 acxaxcyacaxcaxa
42222222 )( acayaxac
)()( 22222222 acayaxac 222 acbLet
222222 bayaxb
12
2
2
2
by
ax standard equation of
a hyperbola
vertextransverse axis
conjugate axis
lactus rectum
length of lactus rectum = ab22
length of the semi-transverse axis = a
length of the semi-conjugate axis = b
asymptote
xaby equation of
asymptote :
Other form of Hyperbola :
12
2
2
2
bx
ay
Rectangular Hyperbola
If b = a, then
222 ayx 12
2
2
2
by
ax
12
2
2
2
bx
ay 222 axy
The hyperbola is said to be rectangular hyperbola.
equation of asymptote : 0yx
If the rectangular hyperbola x2 – y2 = a2 is rotated through 45o about the origin, then the coordinate axes will become the asymptotes.
equation becomes :2
2axy
Simple Parametric Equations and Locus Problems
x = f(t)
y = g(t)parametric equations
parameter
Combine the two parametric equations into one equation which is independent of t. Then sketch the locus of the equation.
Equation of Tangents to Conicsgeneral equation of conics :
022 FEyDxCyBxyAx
Steps :
(1) Differentiate the implicit equation to find .
(2) Put the given contact point (x1,y1) into
to find out the slope of tangent at that point.
(3) Find the equation of the tangent at that point.
dxdy
dxdy
OR
0)(2
)(2
)(2 111111 FyyExxDyCyyxxyBAx
Conics Parabola Ellipse HyperbolaGraph
Definition PF = PN PF1 + PF2 = 2a | PF1 + PF2 | = 2a
Conics Parabola Ellipse HyperbolaGraph
Standard Equation axy 42 12
2
2
2
by
ax 12
2
2
2
by
ax
Conics Parabola Ellipse HyperbolaGraph
Directrix x = -a ,eax ,
eax
PNPFe 1 PN
PFe 1
Conics Parabola Ellipse HyperbolaGraph
Vertices (0,0) A1(a,0), A2(-a,0), B1(0,b), B2(0,-b)
A1(a,0), A2(-a,0)
Conics Parabola Ellipse HyperbolaGraph
Axes axis of parabola = the x-axis
major axis = A1A2
minor axis =B1B2
transverse axis =A1A2
conjugate axis =B1B2
where B1(0,b), B2(0,-b)
Conics Parabola Ellipse HyperbolaGraph
Length of lantus rectum LL’
4aab22
ab22
Conics Parabola Ellipse HyperbolaGraph
Asymptotes ---- ----x
aby
Conics Parabola Ellipse HyperbolaGraph
Parametric representation of P
)2,( 2 atat )sin,cos( ba )tan,sec( ba