SME
Grade 12 Math · Problem type
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.”
Diagram
(x − 2)²/16 + y²/12 = 1
Center (2, 0)
c
=
√
16 − 12
=
2:
the
foci
are
2
from
the
center,
across.
From any point, the distances to the two foci add to the long axis.
This is an example. Type your own values to start a new problem.
r
=
k
1
−
cos
θ
d
Distance from the focus to the directrix
calculated
R
Distance to the vertex at 0°
calculated
S
Distance to the vertex at 180°
calculated
a
Half the major axis
calculated
c
Distance from the center to the focus
worked out (not typed)
b
Half the minor axis
worked out (not typed)
Formulas
k
=
e
×
d
3 = 0.5 × 6
R
=
k
1 −
e
6
=
3
1 − 0.5
S
=
k
1 +
e
2
=
3
1 + 0.5
a
=
R
+
S
2
4
=
6 + 2
2
c
=
a
×
e
2 = 4 × 0.5
b
=
√
R
×
S
3.4641
=
√
6 × 2
Assumptions
•
r = ed ÷ (1 − e cos θ) with 0 < e < 1 is an ellipse with one focus at the pole.
•
Its vertices are at θ = 0° and θ = 180°; the major axis 2a is the sum of their distances.
•
The center is c = ae from the focus, and b² = a² − c².
Step-by-step
WE KNOW
e = 0.5, k = 3
FIND
distance from the focus to the directrix (d), distance to the vertex at 0° (R), distance to the vertex at 180° (S), half the major axis (a), distance from the center to the focus (c), half the minor axis (b)
Step 1 · Find distance from the focus to the directrix (d)
Use k = e × d
The
top
is
ed:
divide
it
by
e
to
find
the
directrix
distance.
d
=
k
e
d
=
3
0.5
d
=
6
Step 2 · Find distance to the vertex at 0° (R)
Use R = k ÷ (1 − e)
At
θ
=
0°,
cos
θ
=
1:
the
bottom
is
1
−
e
.
R
=
3
1 − 0.5
R
=
3
0.5
R
=
6
Step 3 · Find distance to the vertex at 180° (S)
Use S = k ÷ (1 + e)
At
θ
=
180°,
cos
θ
=
−1:
the
bottom
is
1
+
e
.
S
=
3
1 + 0.5
S
=
3
1.5
S
=
2
Step 4 · Find half the major axis (a)
Use a = (R + S) ÷ 2
The
vertices
are
on
opposite
sides
of
the
focus:
the
major
axis
2
a
is
R
+
S
.
a
=
6 + 2
2
a
=
8
2
a
=
4
Step 5 · Find distance from the center to the focus (c)
Use c = a × e
The
center
is
ae
from
the
focus:
e
is
the
share
of
a
between
the
center
and
the
focus.
c
=
4
×
0.5
c
=
2
Step 6 · Find half the minor axis (b)
Use b = √(R × S)
b
2
=
a
2
−
c
2
=
(
a
+
c
)(
a
−
c
),
and
a
+
c
=
R
,
a
−
c
=
S
.
b
=
√
6 × 2
b
=
√
12
b
=
3.4641
Check
Put the numbers back in. Both sides should match.
3 = 0.5 × 6 ✓
6
=
3
1 − 0.5
✓
2
=
3
1 + 0.5
✓
4
=
6 + 2
2
✓
2 = 4 × 0.5 ✓
3.4641
≈
√
6 × 2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Polar equations of conics and rotation of axes
Main lesson
Polar conics with sin θ
Use this for “Name the conic r = 4 ÷ (1 + sin θ), its eccentricity and its directrix.”
A polar parabola
Use this for “Find the vertex and the directrix of r = 4 ÷ (1 − cos θ), and the point at θ = 90°.”
Rotating the axes
Use this for “Through what angle should the axes turn to remove the xy term of 4x² + 2xy + 2y² = 1? Which conic is it?”
The equation after turning the axes
Use this for “Turn the axes to remove the xy term of 4x² + 2xy + 2y² = 1 and write the equation in x′ and y′.”