SME
Grade 12 Math · Problem type
A polar parabola
Use this for “Find the vertex and the directrix of r = 4 ÷ (1 − cos θ), and the point at θ = 90°.”
Diagram
y² = 8(x + 2)
Vertex (−2, 0)
focus (0, 0), p = 2 away
directrix x = −4.
Every point is as far from the focus as from the directrix.
The point (0, 4) is on the curve.
This is an example. Type your own values to start a new problem.
r
=
d
1
−
cos
°
p
Distance from the vertex to the focus
worked out (not typed)
x
x of the point
worked out (not typed)
y
y of the point
worked out (not typed)
Formulas
p
=
d
2
2
=
4
2
r
=
d
1 − cos(
θ
)
4
=
4
1 − cos(90°)
x
=
r
×
cos(
θ
°)
0 = 4 × cos(90°)
y
=
r
×
sin(
θ
°)
4 = 4 × sin(90°)
Assumptions
•
r = d ÷ (1 − cos θ) has e = 1: a parabola with its focus at the pole and directrix x = −d.
•
The vertex is halfway between the focus and the directrix, at θ = 180°: r = d ÷ 2.
Step-by-step
WE KNOW
d = 4, θ = 90°
FIND
distance from the vertex to the focus (p), distance from the pole (r), x of the point (x), y of the point (y)
Step 1 · Find distance from the vertex to the focus (p)
Use p = d ÷ 2
The vertex is halfway from the focus to the directrix.
p
=
4
2
p
=
2
→
the
vertex
is
(−2,
0):
r
=
2
at
θ
=
180°;
the
directrix
is
x
=
−4
Step 2 · Find distance from the pole (r)
Use r = d ÷ (1 − cos(θ))
Put the angle into the equation.
r
=
4
1 − cos(90°)
r
=
4
Step 3 · Find x of the point (x)
Use x = r × cos(θ°)
Across:
r
times
the
cosine
of
the
angle.
x
=
4
×
cos(90°)
x
=
4
×
0
x
=
0
Step 4 · Find y of the point (y)
Use y = r × sin(θ°)
Up:
r
times
the
sine
of
the
angle.
y
=
4
×
sin(90°)
y
=
4
Check
Put the numbers back in. Both sides should match.
2
=
4
2
✓
4
=
4
1 − cos(90°)
✓
0 = 4 × cos(90°) ✓
4 = 4 × sin(90°) ✓
Topic check
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