SME
Grade 12 Math · Precalculus
Polar equations of conics and rotation of axes
Use this for “Name the conic r = 6 ÷ (2 − cos θ), find its eccentricity, and find r at θ = 60°.”
Diagram
Dividing
by
2,
r
=
6
2 − cos θ
is
r
=
3
1 − 0.5 cos θ
,
with
e
=
0.5
and
ed
=
3,
so
d
=
6.
With e < 1 the curve is an ellipse: its focus is the pole and its directrix is x = −6.
The point at θ = 60° has r = 4.
The
point’s
distance
to
the
focus
is
PF
=
4
and
to
the
directrix
PD
=
8,
and
PF
PD
=
0.5,
which
is
e.
This is an example. Type your own values to start a new problem.
r
=
k
−
cos
°
e
Eccentricity
calculated
d
Distance from the focus to the directrix
calculated
Formulas
e
=
|
n
|
÷
m
0.5 = |1| ÷ 2
d
=
k
÷
|
n
|
6 = 6 ÷ |1|
r
=
k
m
−
n
× cos(
θ
)
4
=
6
2 − 1 × cos(60°)
Assumptions
•
Divide the top and bottom by m to reach r = ed ÷ (1 − e cos θ): the eccentricity e and the directrix distance d.
•
e < 1 is an ellipse, e = 1 a parabola, e > 1 a hyperbola; the focus is at the pole.
•
n is the number after the minus: + cos θ is n = −1. With − cos θ the directrix is x = −d; with + cos θ it is x = d.
Step-by-step
WE KNOW
k = 6, m = 2, n = 1, θ = 60°
FIND
eccentricity (e), distance from the focus to the directrix (d), distance from the pole (r)
Step 1 · Find eccentricity (e)
Use e = |n| ÷ m
Dividing
the
bottom
by
m
leaves
1
−
e
cos
θ
,
so
e
is
the
number
before
cos
θ
over
m
.
e
=
|1|
÷
2
e
=
1
2
e
=
0.5
→
an
ellipse
Step 2 · Find distance from the focus to the directrix (d)
Use d = k ÷ |n|
The
top
over
m
is
ed,
and
e
is
|
n
|
÷
m
,
so
d
=
k
÷
|
n
|.
d
=
6
÷
|1|
d
=
6
→
the
directrix
is
x
=
−6
Step 3 · Find distance from the pole (r)
Use r = k ÷ (m − n × cos(θ))
Put the angle into the equation: the point on the conic in that direction.
r
=
6
2 − 1 × cos(60°)
r
=
6
2 − 0.5
r
=
6
1.5
r
=
4
Check
Put the numbers back in. Both sides should match.
0.5 = |1| ÷ 2 ✓
6 = 6 ÷ |1| ✓
4
=
6
2 − 1 × cos(60°)
✓
Topic check
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More problem types
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A polar ellipse: vertices and axes
Use this for “For r = 3 ÷ (1 − 0.5 cos θ), find the vertices, the center and the lengths of the axes.”
A polar parabola
Use this for “Find the vertex and the directrix of r = 4 ÷ (1 − cos θ), and the point at θ = 90°.”
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Use this for “Through what angle should the axes turn to remove the xy term of 4x² + 2xy + 2y² = 1? Which conic is it?”
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Use this for “Turn the axes to remove the xy term of 4x² + 2xy + 2y² = 1 and write the equation in x′ and y′.”