MATH 166H 10TH HOMEWORK
1. Example: The Parabola, Tangent Lines and Pedal Curve
1.1. Tangent Line to a Parabola. A parabola with its focus at the pole and
having as directrix the line r = d sec θ, has the equation
(1)
r=
d
d
= sec2 12 θ
1 + cos θ
2
− π < θ < π.
The tangent line to the parabola at a point A(r1 , θ1 ) has the polar equation
r=
p
cos(θ − θ0 )
with parameters θ0 and p given by
θ0 = θ1 + ϕ,
r1
cos ϕ = p 2
,
r1 + (r10 )2
−r0
sin ϕ = p 2 1 0 ,
r1 + (r1 )2
p = r1 cos ϕ.
We compute from the parabola’s equation (1):
d
sec2 12 θ tan 12 θ,
2
p
d
r2 + (r0 )2 = sec3 12 θ,
2
cos ϕ = cos 21 θ1 ,
r0 =
sin ϕ = − sin 21 θ1 ,
so that
ϕ = − 21 θ1 ,
p=
d
sec 21 θ1 .
2
Figure 1.1 shows a parabola with its tangent at a point A(r1 , θ1 ). The equation of
the tangent line is
p
r=
cos(θ − θ0 )
d
= sec 12 θ1 sec(θ − 12 θ1 ).
2
1
2
MATH 166H 10TH HOMEWORK
Figure 1. Parabola, Directrix, and Tangent Line.
1.2. The Pedal Curve. The point where the pedal line meets the tangent,
d
(p, θ0 ) =
sec 21 θ1 , 12 θ1 ,
2
traces a curve as θ1 varies, called the pedal curve (with respect to the pole). Eliminating the parameter θ1 we see that this curve has the equation
r=
d
sec θ,
2
that is, the pedal curve is the tangent line to the parabola at the vertex; see
Figure 1.2.
2. Problems
2.1. Teams 1, 3 & 5. A logarithmic spiral has the polar equation
r = eaθ
(a = constant > 0).
(1) Find the polar equation of the tangent line to the spiral at a point (r1 , θ1 ).
(2) Show that the angle between the radial line to (r1 , θ1 ) and the radial line
to the pedal point (d, θ0 ) is a constant.
(3) Show that the pedal curve is another logarithmic spiral with the same
parameter a.
MATH 166H 10TH HOMEWORK
3
Figure 2. Pedal curve of the parabola.
2.2. Teams 2, 4 & 6. Consider the circle r = 2a cos θ.
(1) Find the polar equation of the tangent line to the circle at a point (r1 , θ1 ).
(2) Show that the pedal curve is a cardiod with its cusp at the pole.