Yimin Math Centre
4 Unit Math Homework for Year 12
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Table of contents
2
Topic 2 — Conics (Part 2)
1
2.1
Parametric Equations for the Ellipse and Hyperbola . . . . . . . . . . . . . . . . . . .
1
2.1.1
Parametric Equations for the Ellipse . . . . . . . . . . . . . . . . . . . . . . .
1
2.1.2
Parametric Equations for Hyperbola . . . . . . . . . . . . . . . . . . . . . . .
2
2.1.3
Further Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.4
Equation of a Chord on an Ellipse . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.5
Equation of a Chord on Hyperbola . . . . . . . . . . . . . . . . . . . . . . . .
7
Past Exam Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
This edition was printed on June 17, 2013.
Camera ready copy was prepared with the LATEX2e typesetting system.
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4 Unit Math Homework for Year 12
Year 12 Topic 2 Homework
2
Page 1 of 14
Topic 2 — Conics (Part 2)
2.1
2.1.1
Parametric Equations for the Ellipse and Hyperbola
Parametric Equations for the Ellipse
Definition: Let N be the foot of a perpendicular from a point P on the ellipse to the x-axis.
Produce NP to a point Q on the circle x2 + y 2 = a2 .
Let OQ make an angle θ with the positive x-axis, π < θ ≤ π.
Then Q has coordinates (a cos θ, a sin θ).
2
2
At P, xa2 + yb2 = 1 and x = a cos θ, hence y = b sin θ, where a > b.
2
2
Hence the ellipse with Cartesian equation xa2 + yb2 = 1.
has parametric equations x = a cos θ and y = b sin θ, where −π < θ ≤ π.
The circle x2 + y 2 = a2 is called the auxiliary circle.
Exercise 2.1.1 Find the parametric equations of
1. The ellipse
x2
16
+
y2
9
= 1.
2. The ellipse x2 + 4y 2 = 4.
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Year 12 Topic 2 Homework
2.1.2
Page 2 of 14
Parametric Equations for Hyperbola
Definition: N the foot of the perpendicular form a point P on the hyperbola to the x-axis.
From N, a tangent is drawn to the circle x2 + y 2 = a2 , meeting the circle in Q.
If P is on the right-hand branch, choose Q in the same quadrant as P .
If P is on the left-hand branch, choose Q in different quadrants as P .
Let θ be the angle OQ makes with the positive x-axis.
2
2
At P, x = a sec θ and xa2 − yb2 = 1, hence y = b tan θ
For P on the right-hand branch, − π2 < θ < π2 .
For P on the left-hand branch, −π < θ < − π2
2
2
The hyperbola xa2 − yb2 = 1 has parametric equations
x = a sec θ and y = b tan θ, where −π < θ ≤ π, θ 6= ± π2 .
Exercise 2.1.2 Find the parametric equations of
1. The hyperbola
x2
16
−
y2
25
= 1.
2. The hyperbola x2 − y 2 = 4.
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Year 12 Topic 2 Homework
Exercise 2.1.3 Find the Cartesian equations of:
1. The ellipse x = 3 cos θ, y = 2 sin θ.
2. The ellipse x = 5 cos θ, y = 4 sin θ.
3. The hyperbola x = 3 sec θ, y = 4 tan θ.
4. The hyperbola x = 2 sec θ, y = 5 tan θ.
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Page 3 of 14
Year 12 Topic 2 Homework
2.1.3
Page 4 of 14
Further Trigonometry
Sums and Differences:
sin(θ + φ) = sin θ cos φ + cos θ sin φ
sin(θ − φ) = sin θ cos φ − cos θ sin φ
cos(θ + φ) = cos θ cos φ − sin θ sin φ
cos(θ − φ) = cos θ cos φ + sin θ sin φ
tan(θ + φ) =
tan θ + tan φ
1 − tan θ tan φ
tan(θ − φ) =
tan θ − tan φ
1 + tan θ tan φ
Double Angle Identities:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1
2 tan θ
tan 2θ =
1 − tan2 θ
Sums to Products:
sin θ + sin φ = 2 sin
θ+φ
2
cos
θ−φ
sin θ − sin φ = 2 cos
sin
2
θ+φ
θ−φ
cos θ + cos φ = 2 cos
cos
2
2
θ+φ
θ−φ
cos θ − cos φ = 2 sin
sin
2
2
θ+φ
2
θ−φ
2
The t-Formulae: Let t = tan 2θ , Then:
sin θ =
2t
,
1 + t2
cos θ =
1 − t2
,
1 + t2
tan θ =
2t
1 − t2
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Year 12 Topic 2 Homework
2.1.4
Page 5 of 14
Equation of a Chord on an Ellipse
PQ, PR, PW are chords of the conic.
Chord PR through the centre is a diameter,
PW through the focus S is a focal chord.
Cd is a focal chord which is perpendicular to the major axis.
Such a chord is termed a latus rectum of the conic and has equation x = ±ae.
The equation of a general chord PQ is best expressed in terms of the parameters
θ and φ of the end points P and Q.
2
y2
Definition: Let P (a cos θ, b sin θ) and Q(a cos φ, b sin φ) lie on xa2 + b2 = 1.
Then the gradient of PQ is:
2 sin( θ−φ
) cos(θ+φ)
cos( θ+φ
)
b(sin θ−sin φ)
b
2
2
= ab × −2 sin( θ−φ
×
=
−
θ+φ
a(cos θ−cos φ)
a
) sin(
)
sin( θ+φ )
2
cos( θ+φ )
2
2
2
∴ y − b sin θ = − ab × sin( θ+φ
(x − a cos θ)
)
2
θ+φ
y
θ+φ
θ+φ
x
cos
+
sin
=
cos
θ
cos
+ sin θ sin
a
2
b
2
2
θ+φ
x
Then the equation of the chord is: a cos 2 + yb sin
θ+φ
2
θ+φ
)
2
= cos( θ−φ
.
2
In the special case where PQ is a diameter, (0, 0) lies on the chord and hence |θ − φ| = π
Example 2.1.1 P (a cos θ, b sin θ) and Q[a cos(π + θ), b sin(π + θ)] lie on the ellipse
Show that P Q passes through (0, 0).
2
2
Solution: The equation of the chord PQ of the ellipse xa2 + yb2 = 1 is:
y
x
+ b sin θ+φ
= cos θ−φ
, where P, Q have parameters θ, φ
cos θ+φ
a
2
2
2
We have
of the chord
φ = π+ θ. Hence
the equation
PQ transforms into:
θ+(π+θ)
θ+(π+θ)
θ−(π+θ)
y
x
cos
+ b sin
= cos
.
a
2
2
2
x
cos 2θ+π
+ yb sin 2θ+π
= cos −π
.
a
2
2
2
y
x
Thus − a sin θ + b cos θ = 0. Therefore (0, 0) lies on the chord PQ.
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x2
a2
+
y2
b2
= 1.
Year 12 Topic 2 Homework
Page 6 of 14
Exercise 2.1.4 The point P (a cos θ, b sin θ), and Q(a cos φ, b sin φ) lie on the ellipse
2
and the chord PQ subtends a right angle at (0, 0). Show that tan θ tan φ = − ab2 .
x2
a2
+
y2
b2
=1
Exercise 2.1.5 The point P (a cos θ, b sin θ) and Q[a cos(−θ), b sin(−θ)] are the extremities of the
latus
2
2
rectum x = ae of the ellipse xa2 + yb2 = 1.
1. Show that cos θ = e.
2. Show that PQ has length
2b2
.
a
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Year 12 Topic 2 Homework
2.1.5
Page 7 of 14
Equation of a Chord on Hyperbola
2
2
Definition: Let P (a sec θ, b tan θ) and Q(a sec φ, b tan φ) lie on xa2 − yb2 = 1.
y
Then the equation of the chord is: xa cos θ−φ
) = cos( θ+φ
− b sin θ+φ
.
2
2
2
In the special case where PQ is a diameter, (0, 0) lies on the chord and hence |θ + φ| = π
2
2
Example 2.1.2 The points P (a sec θ, b tan θ) and Q(a sec φ, b tan φ) lie on the hyperbola xa2 − yb2 = 1
2
and the chord PQ subtends a right angle at (0, 0). Show that sin θ sin φ = − ab2 .
Solution: P OQ is a right-angled triangle. Therefore OP 2 + OQ2 = P Q2 .
a2 sec2 θ + b2 tan2 θ + a2 sec2 φ + b2 tan2 φ = a2 (sec θ − sec φ)2 + b2 (tan θ − tan φ)2
= a2 (sec2 θ − 2 sec θ sec φ + sec2 φ)
+ b2 (tan2 θ − 2 tan θ tan φ + tan2 φ)
Then 0 = −2a2 sec θ sec φ − 2b2 tan θ tan φ, ⇒ ∴ sin θ sin φ = −
a2
b2
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Year 12 Topic 2 Homework
Exercise 2.1.6 P (a sec θ, b tan θ) lie on the hyperbola
S 0 (−ae, 0).
Page 8 of 14
x2
a2
−
y2
b2
= 1 with foci S(ae, 0)
1. Show that P S = a(e sec θ − 1) and P S 0 = a(e sec θ = 1)
2. Deduce that |P S − P S 0 | = 2a.
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and
Year 12 Topic 2 Homework
Page 9 of 14
Exercise 2.1.7 The point P (a cos θ, b sin θ) and Q(a cos φ, b sin φ) lie on the ellipse
2
If P Q subtends a right angle at (a, 0) show that tan 2θ tan φ2 = − ab 2 .
x2
a2
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+
y2
b2
= 1.
Year 12 Topic 2 Homework
Page 10 of 14
Exercise 2.1.8 The points P (a sec θ, b tan θ) and Q(a sec φ, b tan φ) lie on the hyperbola
2
2
x2
− yb2 = 1. If PQ subtends a right angle at (a, 0). Show that tan 2θ tan φ2 = − ab 2 .
a2
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Year 12 Topic 2 Homework
Page 11 of 14
Exercise 2.1.9
2
2
1. Given that the equation of the chord PQ of the ellipse xa2 + yb2 = 1 is xa cos( θ+φ
) + yb sin( θ+φ
)=
2
2
θ−φ
θ
cos( 2 ), where P, Q have parameters θ, φ, show that if PQ is a focal chord then tan 2 tan φ2
1−e
takes one of values − 1+e
or 1+e
.
1−e
2. PQ is a focal chord of
x2
4
+
y2
3
= 1, where P has coordinates (1, 32 ). Find the coordinate of Q.
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Year 12 Topic 2 Homework
Page 12 of 14
Exercise 2.1.10
2
2
1. P (a sec θ, b tan θ) and Q(a sec φ, b tan φ), lie on the hyperbola xa2 − yb2 = 1. Use the result that
) − yb sin( θ+φ
) = cos( θ+φ
) to show that if PQ is a focal
the chord PQ has equation xa cos( θ−φ
2
2
2
φ
θ
1−e
1+e
chord, then tan 2 tan 2 takes one of the values 1+e or 1−e .
√
√
2
2
2. P (2 3, 3 3) is one extremity of a focal chord on the hyperbola x3 − y9 = 1. Find the coordinates
of the other extremity Q.
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Year 12 Topic 2 Homework
Page 13 of 14
2
2
Exercise 2.1.11 Points P (a cos θ, b sin θ and Q(a cos φ, bsinφ) lie on the ellipse xa2 + yb2 = 1. Find
the equation of the chord PQ. Hence show that if PQ subtends a right angle at the point A(a, 0)
then PQ passes through a fixed point on the x-axis.
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Year 12 Topic 2 Homework
2.2
Past Exam Questions
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Page 14 of 14