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The University of British Columbia
MATH 308, Section 101, Instructor Tai-Peng Tsai
Final Exam – December 2011
Family Name Given Name
Student Number Signature
No notes nor calculators.
Rules Governing Formal Examinations:
- Each candidate must be prepared to produce, upon request, a
UBCcard for identification;
- Candidates are not permitted to ask questions of the invigila-
tors, except in cases of supposed errors or ambiguities in exami-
nation questions;
- No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination;
- Candidates suspected of any of the following, or similar, dishon-
est practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action;
(a) Having at the place of writing any books, papers or
memoranda, calculators, computers, sound or image play-
ers/recorders/transmitters (including telephones), or other mem-
ory aid devices, other than those authorized by the examiners;
(b) Speaking or communicating with other candidates; (c) Purposely exposing written papers to the view of other can-
didates or imaging devices. The plea of accident or forgetfulness
shall not be received;
- Candidates must not destroy or mutilate any examination ma-
terial; must hand in all examination papers, and must not take
any examination material from the examination room without
permission of the invigilator; and
- Candidates must follow any additional examination rules or
directions communicated by the instructor or invigilator.
problem max score
total 100
- Suppose a conic has two focus points F (4, 0) and F ′(− 4 , 0).
(5 pt) (a)Find the equation of the conic if a directrix has the equation x = 9.
(5 pt) (b)Find the equation of the conic if the conic has eccentricity e = 2.
- The triangle 4 ABC has vertices A(− 1 , 2), B(− 3 , −1) and C(3, 1), and the points P (1, 13 ),
Q(1,
3 2 ), and^ R(−^
5 3 ,^ 1) lie on^ BC,^ CA^ and^ AB, respectively.
(7 pt) (a)Determine the ratios in which P , Q and R divide the sides of the triangle.
(3 pt) (b)Determine whether or not the lines AP , BQ and CR are concurrent.
(5 pt) 4. (a)Find the Euclidean transformation which is the counterclockwise rotation about the
point (2, 2) by angle
π
3
(5 pt) (b)Find the affine transformation which maps the points
and
to the points ( 1
− 1
and
, respectively.
(5 pt) 5. (a)Find the image of the line 3x + y = 4 under the affine transformation t(x) = Ax +
with A =
(5 pt) (b)Find the image of the circle |z| = 1 under the M¨obius transformation M (z) =
iz + 1
z + i
(10 pt) 6. Let two circles S 1 and S 2 intersect in A and B, and let the diameters of S 1 and S 2 through B
cut S 2 and S 1 in C and D. Let S 3 be the circle containing B, C, and D. Show that line
AB
passes through the center of circle S 3.
(7 pt) 8. (a)Find the d-line that passes through the two d-points i/3 and − 1 /3, and sketch it.
(8 pt) (b)Find the Euclidean radius of the non-Euclidean circle C = {z ∈ D : d(
i
3
, z) =
Do not evaluate or simplify the solution. Note ln 2 ≈ 0 .6931 and ln 3 ≈ 1 .0986.
(10 pt) 9. Answer YES or NO to the following questions. Anything else will not earn any partial credits.
Answering both YES and NO earns 0.
(a) Is it true that an affine transformation preserves ratio of lengths on parallel lines?
(b) Is it true that, for any point P inside triangle 4 ABC, there is an affine transformation t
such that t(P ) is outside of triangle 4 t(A)t(B)t(C)?
(c) Is it true that an ellipse can be mapped onto a circle by an affine transformation?
(d) Is it true that an ellipse can be mapped onto a circle by an inversive transformation?
(e) Is it true that an inversion is a M¨obius transformation?
