Problems in Geometry
(7)
1.
Let
ϕ
be an ellipse with foci
F, F
0
. If the tangent lines at
A, B
∈
ϕ
intersect in
P ,
prove that
PF
⊥
AB
iff
F
∈
AB .
1
2.
Let
ϕ
be an ellipse with foci
F, F
0
. Let
t, t
0
be the tangents to
ϕ
which intersect in
P .
If
H, H
0
are respectively the feet of the perpendiculars from
F
on
t, t
0
,
prove that
PF
0
is
perpendicular to
HH
0
.
2
3.
Let
ϕ
be an ellipse of foci
F, F
0
. Let
ϕ
∩
FF
0
=
{
A, A
0
}
.
Let
‘, ‘
0
be the tangents to
ϕ.
at
A, A
0
.
For any tangent
t
to
ϕ
at
M
∈
ϕ
let
t
∩
‘
=
{
P
}
, t
∩
‘
0
=
{
P
0
}
.
(A) Prove that
<
(
FP, FP
0
) =
<
(
F
0
P, F
0
P
0
) =
π/
2
3
.
(B) Prove that
FP, F
0
P
0
intersect on the normal to
ϕ
at
M.
4
4.
Let
ϕ
be an ellipse of foci
F, F
0
.
Consider
X
∈
ϕ.
Let
Q, Q
0
be the points in which the
normal of
ϕ
at
X
intersects the perpendiculars to
XF, XF
0
erected at
F, F
0
respectively.
Prove that, the perpendicular bisector of [
FF
0
] bisects [
Q, Q
0
]
.
5.
Consider an ellipse
ϕ
with foci
F, F
0
. Let a line through
F
meet
ϕ
in
X, X
0
.
If the
normals to
ϕ
at
X, X
0
intersect in
N,
prove that the parallel to
FF
0
through
N
bisects
[
XX
0
]
.
5
6.
(A) Let
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 Spring '11
 cemtezer
 Geometry, Conic section, Poncelet

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