SME
Grade 12 Math · Problem type
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?”
Diagram
The
turn
has
tan
2θ
=
2
4 − 2
=
1,
so
2θ
=
45°
and
θ
=
22.5°.
In the turned axes the equation reads 4.41x′² + 1.59y′² = 1, with no x′y′ term: an ellipse, as B² − 4AC = −28 < 0 says.
Its
half-axes
are
√(
1
A′
)
=
0.48
along
x′
and
√(
1
C′
)
=
0.79
along
y′.
This is an example. Type your own values to start a new problem.
A
x²
+
B
xy
+
C
y²
+
Dx
+
Ey
+
F
=
0
Δ
Discriminant B² − 4AC
worked out (not typed)
θ
Angle to turn the axes
worked out (not typed) · °
Formulas
Δ
=
B
2
−
4
×
A
×
C
−28
=
2
2
−
4
×
4
×
2
tan(2
×
θ
)
=
B
A
−
C
tan(2
×
22.5°)
=
2
4 − 2
Assumptions
•
In Ax² + Bxy + Cy² + Dx + Ey + F = 0, turning the axes by θ with cot 2θ = (A − C) ÷ B removes the xy term.
•
Take 2θ from 0° to 180°, so θ is from 0° to 90°; A = C gives θ = 45°.
•
B² − 4AC names the conic: below 0 an ellipse, 0 a parabola, above 0 a hyperbola.
Step-by-step
WE KNOW
A = 4, B = 2, C = 2
FIND
discriminant B² − 4AC (Δ), angle to turn the axes (θ)
Step 1 · Find discriminant B² − 4AC (Δ)
Use Δ = B² − 4 × A × C
Turning
the
axes
changes
A
,
B
and
C
but
not
B
2
−
4AC,
so
it
names
the
conic.
Δ
=
2
2
−
4
×
4
×
2
Δ
=
4
−
4
×
4
×
2
Δ
=
4
−
16
×
2
Δ
=
4
−
32
Δ
=
−28
→
an
ellipse
Step 2 · Find angle to turn the axes (θ)
Use tan(2 × θ) = B ÷ (A − C)
tan
2
θ
=
B
A
−
C
is
cot
2
θ
=
A
−
C
B
turned
over;
add
180°
to
a
negative
2
θ
,
then
halve.
θ
=
tan⁻¹
B
÷ (
A
−
C
)
2
θ
=
tan⁻¹
2 ÷ (4 − 2)
2
θ
=
45°
÷
2
θ
=
22.5°
Check
Put the numbers back in. Both sides should match.
−28
=
2
2
−
4
×
4
×
2
✓
tan(2
×
22.5°)
=
2
4 − 2
✓
Topic check
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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 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°.”
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′.”