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ISI B.Stat Entrance Exam
2010 ISI B.Stat Entrance Exam
9
9
Part of
2010 ISI B.Stat Entrance Exam
Problems
(1)
d(A,B)=d(f(A),f(B))
Source: ISI(BS) 2010 #9
5/16/2012
Let
f
:
R
2
→
R
2
f: \mathbb{R}^2 \to \mathbb{R}^2
f
:
R
2
→
R
2
be a function having the following property: For any two points
A
A
A
and
B
B
B
in
R
2
\mathbb{R}^2
R
2
, the distance between
A
A
A
and
B
B
B
is the same as the distance between the points
f
(
A
)
f(A)
f
(
A
)
and
f
(
B
)
f(B)
f
(
B
)
.Denote the unique straight line passing through
A
A
A
and
B
B
B
by
l
(
A
,
B
)
l(A,B)
l
(
A
,
B
)
(a) Suppose that
C
,
D
C,D
C
,
D
are two fixed points in
R
2
\mathbb{R}^2
R
2
. If
X
X
X
is a point on the line
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
, then show that
f
(
X
)
f(X)
f
(
X
)
is a point on the line
l
(
f
(
C
)
,
f
(
D
)
)
l(f(C),f(D))
l
(
f
(
C
)
,
f
(
D
))
.(b) Consider two more point
E
E
E
and
F
F
F
in
R
2
\mathbb{R}^2
R
2
and suppose that
l
(
E
,
F
)
l(E,F)
l
(
E
,
F
)
intersects
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
at an angle
α
\alpha
α
. Show that
l
(
f
(
C
)
,
f
(
D
)
)
l(f(C),f(D))
l
(
f
(
C
)
,
f
(
D
))
intersects
l
(
f
(
E
)
,
f
(
F
)
)
l(f(E),f(F))
l
(
f
(
E
)
,
f
(
F
))
at an angle
α
\alpha
α
. What happens if the two lines
l
(
C
,
D
)
l(C,D)
l
(
C
,
D
)
and
l
(
E
,
F
)
l(E,F)
l
(
E
,
F
)
do not intersect? Justify your answer.
function
geometry