Conic Sections Imagine you slice through a cone at different ang circle ellipse parabola rectangular hyperbola You could get a cross-section which is These shapes are all important functions in Mathematics and occur in fields as diverse as the motion of planets to the optimum design of a satellite dish. In FP1 you consider the algebra & geometry of 2 of these – the parabola and rectangular hyperbola
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Conic Sections Imagine you slice through a cone at different angles circle ellipse parabola rectangular hyperbola You could get a cross-section which is.
Sub in Figure 1 WB14 Figure 1 shows a sketch of the parabola C with equation (a) The point S is the focus of C. Find the coordinates of S. (b) Write down the equation of the directrix of C. Figure 1 shows the point P which lies on C, where y > 0, and the point Q which lies on the directrix of C. The line segment QP is parallel to the x-axis. (c) Given that the distance PS is 25, write down the distance QP, (d) find the coordinates of P, (e) find the area of the trapezium OSPQ. where the focus is S( a,0) and the directrix has equation x = - a Coordinates of S are (9,0) Equation of directrix x = -9 Focus-directrix property: PS = PQ QP = 25 Coordinates of P are (16,24)
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Conic SectionsImagine you slice through a cone at different angles
circle
ellipse
parabola
rectangular hyperbola
You could get a cross-section which is a:
These shapes are all important functions in Mathematics and occur in fields as diverse as the motion of planets to the optimum design of a satellite dish.
In FP1 you consider the algebra & geometry of 2 of these – the parabola and rectangular hyperbola
Parabola.aggThe Parabola
Q is the point horizontally in line horizontally with P on the line x = -a
The restriction that P can move such that QP = PS is the focus-directrix property
The locus of points for P is a parabolaPQ
S(a,0)
x = -a
P can move such that QP=PS …
Consider a point P that can move according to a rule:
The point S has coordinates (a,0)
The point S(a,0) is called the focus
The line x = -a is called the directrix
The Cartesian equation is y2 = 4ax
Sub in
Figure 1
WB14 Figure 1 shows a sketch of the parabola C with equation (a) The point S is the focus of C. Find the coordinates of S.
(b) Write down the equation of the directrix of C.
Figure 1 shows the point P which lies on C, where y > 0, and the point Q which lies on the directrix of C. The line segment QP is parallel to the x-axis.(c) Given that the distance PS is 25, write down the distance QP,
(d) find the coordinates of P,
(e) find the area of the trapezium OSPQ.
xy 362
axy 42 where the focus is S(a,0)and the directrix has equation x = -a
Coordinates of S are (9,0)
Equation of directrix x = -9
Focus-directrix property: PS = PQ
QP = 25
259 16
16x xy 362 24 yCoordinates of P are (16,24)
24
242259Area
9
408
WB15 Figure 1 shows a sketch of part of the parabola with equation y2 = 12x .The point P on the parabola has x-coordinate 3
1
The points A and B lie on the directrix of the parabola.The point A is on the x-axis and the y-coordinate of B is positive.Given that ABPS is a trapezium,(b) calculate the perimeter of ABPS.
Figure 1The point S is the focus of the parabola.(a) Write down the coordinates of S.
axy 42 where the focus is S(a,0)
Directrix has equation x = -a
Coordinates of S are (3,0)3
Sub in 31x xy 122 2 y
3
313
2
313
Focus-directrix property
313 PBPS
2 AB3 OSAO
Perimeter = 3214
at P
Eg a curve has parametric equations ,
Parametric functions
Some simple-looking curves are hard to describe with a Cartesian equation.Parametric equations, where the values of x and y are determined by a 3rd
variable t, can be used to produce some intricate curves with simple equations.
tx 2 2ty
t -3 -2 -1 0 1 2 3
x
y
-6
Complete the table and sketch the curve
9
-4
4
-2
1
0
0
2
1
4
4
6
9
NB: you can still find the Cartesian equation of a function defined parametrically…
tx 22xt
Sub in 2ty 2
2
xy 2
41 x
Problem solving with parametric functionsEg a curve has parametric equations ,1tx 24 ty The curve meets the x-axis at A and B, find their coordinates
A B
At A and B, 0y 42 t
2 t
13, x
Coordinates are (-3,0) and (1,0)
Eg a curve has parametric equations ,2tx ty 4The line meets the curve at A. Find the coordinates of A04 yx
0442 tt
02 2 t2 t
84 ,A
Substitute the expressions for x and y in terms of t to solve the equations simultaneously
Solve
Substitute value of t back into expressions for x and y
Find values of t at A and B
Substitute values of t back into expression for x
The parametric form of a parabola is , 2atx aty 2 Does this fit with its Cartesian equation?
Sub into axy 42 22 42 ataat 2222 44 tata which is true!
Exam questions sometimes involve the parabola’s parametric form…
WB16 The parabola C has equation y2 = 20x.(a) Verify that the point P(5t2 ,10t) is a general point on C.
The point A on C has parameter t = 4.The line l passes through A and also passes through the focus of C.(b) Find the gradient of l.
Sub in tt 105 2 , xy 202 22 52010 tt
4t 4080,A
axy 42 has focus S(a,0)
05,S5 axy 202
05,S
4080,A
75
40
7540 of Gradient l 15
8l
22 100100 tt
The equation of the straight line with gradient m that passes through is
(b) Show that the equation of the tangent to C at P(12t2, 24t) is x − ty + 12t2 = 0.
The tangent to C at the point (3, 12) meets the directrix of C at the point X.(c) Find the coordinates of X
WB17 The parabola C has equation y2 = 48x.The point P(12t2, 24t) is a general point on C.(a) Find the equation of the directrix of C.
axy 42 where the focus is S(a,0)and the directrix has equation x = -a
Equation of directrix x = -12
Sub 212tx t1 at P212
32tdx
dy
Giving tangent
)( 11 xxmyy ),( 11 yx
)( 21 1224 txty t 22 1224 txtty
012 2 ttyx
123,
X
Comparing (3,12) with (12t2, 24t) 21 t at (3,12)
Sub n equation of tangent 21t 032
1 yxWhen this intersects directrix x = -12
0312 21 y 18 y
Coordinates of X are (-12,-18)
12xDirectrix
xy 482 21
34 xy 21
32 xdxdy
x32
The Rectangular Hyperbola
The rectangular hyperbola also has a focus-directrix property, but it is beyond the scope of FP1. You only need to know that:
The Cartesian equation is xy = c2
The parametric form of a parabola is , ctx tcy
Problems involving rectangular hyperbola usually require to find the equation of the tangent or normal
for functions given explicitly or in terms of c
Sub ctx 2
1t
22
2
tcc
dxdy
2cxy 12 xcy 22 xcdxdy
2
2
xc
WB19 The rectangular hyperbola H has equation xy = c2, where c is a positive constant. The point A on H has x-coordinate 3c.(a) Write down the y-coordinate of A.
(b) Show that an equation of the normal to H at A is
2cxy with general point tcct,
3 t 3 coordinate cy
Sub91
at A 22
3cc
dxdy
33 cc,
9 normal of Gradient The equation of the straight line with gradient m that passes through is
Giving normal
)( 11 xxmyy ),( 11 yx
)( cxy c 393
cxcy 81273 cxy 80273
cxy 80273
(c) The normal to H at A meets H again at the point B. Find, in terms of c, the coordinates of B.
2cxy xcy 2
Sub in cxy 80273 cxx
c 802723
Solve and simultaneously to find points of intersection
cxxc 80273 22
038027 22 ccxx
0 3 cx x27 c
27cx at B
Given solution x = 3c
cy 27
Coordinates of B are cc 27 27 ,
2cxy cxy 80273
using xcy 2
2cxy 12 xcy 22 xcdxdy
2
2
xc
WB20 The point P , t ≠ 0, lies on the tt 66 ,
(a) Show that an equation for the tangent to H at P is
(b) The tangent to H at the point A and the tangent to H at the point B meet at the point (−9, 12). Find the coordinates of A and B.
rectangular hyperbola H with equation xy = 36.
ttxy 121
2
The equation of the straight line with gradient m that passes through is
Sub 2
1t
at Ptdx
dy t
6
6
Giving tangent
)( 11 xxmyy ),( 11 yx
)( txytt 6216
ttt xy 6162
tt 66 ,
ttxy 121
2
Sub in 129, ttxy 121
2
tt129
212
tt 12912 2
0344 2 tt03624 2 ttt
0123122 ttt 01232 tt
21
23 , t
Sub in 63 49 ,,, tt 66 ,
2cxy 12 xcy 22 xcdxdy
2
2
xc
WB18 The rectangular hyperbola H has equation xy = c2, where c is a constant.
tcct,
(a) Show that the tangent to H at P has equation t2y + x = 2ct.
The point P is a general point on H.
The tangents to H at the points A and B meet at the point (15c, –c).(b) Find, in terms of c, the coordinates of A and B.
The equation of the straight line with gradient m that passes through is