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The University of British Columbia
MATH 308, Section 101, Instructor Tai-Peng Tsai
Final Exam – December 2010
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total
100
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December 2010
Math 308 Section 101 Final Exam
(10 points) 1. Find the two pairs of focuses and directrices of the ellipse
x2 + 4y 2 + 2x = 0.
Page 2 of 12
December 2010
Math 308 Section 101 Final Exam
Page 3 of 12
(10 points) 2. Let a > 0, b > 0, and F and G be the parabolas y 2 = 4a(x + a) and y 2 = 4b(−x + b). The
origin is the focus of both parabolas. Suppose F meets G at P above the x-axis. Use the
reflection property of parabolas to prove that F and G cross at P at right angle.
December 2010
Math 308 Section 101 Final Exam
Page 4 of 12
3. Classify the conics in R2 with the following equations. You do not need to find their axes.
(5 points)
(a) 3x2 − 8xy + 2y 2 − 2x + 4y − 16 = 0
(5 points)
(b) x2 + 8xy + 16y 2 − x + 8y − 12 = 0
December 2010
Math 308 Section 101 Final Exam
Page 5 of 12
(10 points) 4. On a triangle ABC, D and D0 are points on side BC so that ∠BAD = ∠D0 AC, E and E 0 are
points on side CA so that ∠CBE = ∠E 0 BA, and F and F 0 are points on sides AB so that
∠ACF = ∠F 0 CB. Suppose that AD, BE, CF are concurrent. Show that AD0 , BE 0 , CF 0 are
BD
AB sin ∠BAD
also concurrent. Hint: You may use
=
.
DC
AC sin ∠DAC
December 2010
Math 308 Section 101 Final Exam
(10 points) 5. Find an affine transformation which maps the ellipse
(It is not unique, but you only need to find one.)
x2 + 4y 2 + 2x = 0
Page 6 of 12
to the unit circle.
December 2010
Math 308 Section 101 Final Exam
(5 points) 6. (a) Find the affine transformation which maps the points
8
3
0
points
,
and
, respectively.
13
4
−1
(5 points)
Page 7 of 12
1
2
3
,
and
to the
−1
−2
−4
(b) Find the projective transformation which maps the Points [1, 0, 0], [0, 1, 0], [0, 0, 1] and
[1, 1, 1] to the Points [2, 1, 0], [1, 0, −1], [0, 3, −1] and [3, −1, 2].
December 2010
(5 points)
Math 308 Section 101 Final Exam
Page 8 of 12
(c) Find the Point of intersection of the Lines L1 : x + 2y + 5z = 0 and L2 : 3x − y + z = 0 .
7. Let A = [2, 1, 1], B = [−1, 1, −1], C = [1, 2, 0] and D = [−1, 4, −2] be four points in RP2 .
(5 points)
(a) Verify that A, B, C, D are collinear and find the equation of the Line.
December 2010
Math 308 Section 101 Final Exam
(5 points)
(b) Find the cross ratio (ABCD).
(5 points)
(c) Find a point X in the same line so that (ABCX) = −1.
Page 9 of 12
December 2010
Math 308 Section 101 Final Exam
Page 10 of 12
(10 points) 8. Find the imagies of the circles |z| = 1 and |z| = 2 under the Möbius transformation
z−2
t(z) =
.
z+i
December 2010
Math 308 Section 101 Final Exam
Page 11 of 12
(10 points) 9. Let C be a circle with radius r and centered at O. For any ordinary point P 6= O, let P 0 be
the inversion of P in C. Then the line p through P 0 and perpendicular to OP P 0 is called the
polar of P for the circle C.
Show that, if the polar of P passes through a point Q, then the polar of Q passes through P .
December 2010
Math 308 Section 101 Final Exam
Page 12 of 12
Some Formulas
eccentricity
ellipse
0≤e<1
hyperbola
e>1
parabola
e=1
standard eq
x2 y 2
+ 2 =1
a2
b
x2 y 2
− 2 =1
a2
b
focus
directrix
(±ae, 0)
x = ±a/e
(±ae, 0)
x = ±a/e
(a, 0)
x = −a
y 2 = 4ax
tangent at (x1 , y1 )
x 1 x y1 y
+ 2 =1
a2
b
x 1 x y1 y
− 2 =1
a2
b
y1 y = 2a(x + x1 )
p
Above b = a |1 − e2 |
1. ellipsoid
x2 y 2 z 2
+ 2 + 2 =1
a2
b
c
2. hyperboloids of one and two sheets, elliptic cone
x2 y 2 z 2
+ 2 − 2 = 1, −1, 0
a2
b
c
3. elliptic paraboloid and hyperbolic paraboloid
z=
x2 y 2
± 2
a2
b
4. The affine transformation of R2 mapping [ 00 ], [ 10 ] and [ 01 ] to P = [ pp12 ], Q = [ qq12 ], and
R = [ rr12 ] is
q
−
p
r
−
p
1
1
1
1
t(~x) = P~ + A~x, A = (P~Q|P~R) =
.
q2 − p2 r2 − p2
5. Affine transformations preserve collinearity, coincidence, parallel lines, and signed ratio of
lengths along the same direction.
6. Ceva’s and Menelaus’ theorems: P ∈ BC, Q ∈ CA, R ∈ AB,
coincidence or collinearity.
AR
RB
·
BP
PC
·
CQ
QA
= ±1 implies
7. The projective transformation of RP2 mapping [1, 0, 0], [0, 1, 0], [0, 0, 1] and [1, 1, 1] to
P = [p1 , p2 , p3 ], Q = [q1 , q2 , q3 ], R = [r1 , r2 , r2 ] and S = [s1 , s2 , s3 ] is
hui
t([~x]) = [A~x], A = [uP | |vQ| |wR| ], where [P | |Q| |R| ] wv = S | .
Projective transformations preserve collinearity, coincidence, and cross ratio.
8. The Line passing Points P and Q has equation det(X | |P | |Q| ) = 0.
9. If A = [a], B = [b], C = [c], D = [d] are 4 collinear points with c = αa + βb and d = γa + δb,
then (ABCD) = (β/α)/(δ/γ) = (AC/CB)/(AD/DB). If (ABCD) = k, then
(BACD) = (ABDC) = 1/k and (ACBD) = (DBCA) = 1 − k.
p
10. Inversion in the unit circle: (x, y) 7→ ( rx2 , ry2 ) where r = x2 + y 2 . Inversion in the circle
R2
|z − c| = R is z 7→ z̄−c̄
+ c.
11. The stereographic projection π : S2 → C satisfies π : (X, Y, Z) 7→
π −1 : x + iy 7→ x2 +y1 2 +1 (2x, 2y, x2 + y 2 − 1).
X+iY
1−Z
with