92. CA , CB tangent to parabola - OIMU 2008 Problem 4.
by Virgil Nicula, Aug 28, 2010, 3:22 PM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=364018
PP1. 
of 
belong to parabola 
with equation 
, 
and 
, 
are tangent to . Find 
, where 
is length of 
-median.
Preliminary. We know that
for any point 
of parabola , where 
. For is well-known the vertex 
, the focus
, i.e. 
and the director 
with equation 
so that for any point 
there is the definition relation
. Thus, 
, i.e. 
.
Proof 1 (analytic). Suppose w.l.o.g. that equation of parabola is 
, where vertex is 
, focus is 
and equation of director is 
. Suppose that
. 
is intersection of tangents in 
, 
to parabola and which have equations 
. Thus, 
. The
midpoint of![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
is 
and 
. Thus, 
and 
. The area
![$S=[ABC]$](//latex.artofproblemsolving.com/b/3/a/b3ae3d445111e4dd28be75922309d3270079368c.png)
is 
. With invariant transformations
get 
.
Proof 2 (shorter, synthetic-analytic).
. Observe that 
and 
, i.e. 
. Therefore, the area 
of is 
a.s.o.
Propose four similar problems which use the remarkable relations between the points 
and 
.[/size]
AP1 Let be the parabola with equation , where 
. Consider a fixed point 
and two mobile points 
so that
. Find the locus of 
, where mean by 
- the tangent in the point 
to the parabola . Study some particular cases.
AP2 Let be the parabola with equation , where . Consider a fixed circle 
, where 
and two mobile points
so that
is tangent to 
. Find the locus of , where mean by - the tangent in the point to the parabola . Study some
particular cases. What is the locus of
if instead of the circle have the circle 
? What happen in the general case of some circle ?
AP3 Let 
, 
be three parabolas with equations 
, 
where 
and 
. Consider two mobile
points
so that the line 
is tangent to 
. Prove that the intersection of tangents to 
in the points 
, 
belongs to 
.
AP4 Given are the parabola 
and the circle 
, where 
. The tangent in a mobile point 
which belongs to the circle cut the
parabola in the points
, 
. Denote the intersection 
of the tangents in the points , to the parabola. Ascertain the geometrical locus of the point .
PP2. Let 
be a function, where 
and 
. Let be a line so that 
.
For any the tangent to 
in the point 
cut again in the point 
. Prove that the points 
are collinearly.
Proof. Let
where . Thus, 
![$\left(x_2-x_1\right)\left(x_3-x_1\right)\left(x_3-x_2\right)\left[m\left(x_1+x_2+x_3\right)+n\right]=0\iff$](//latex.artofproblemsolving.com/1/6/a/16a0196103c91b80c6ee64ef400396181d6783d6.png)
. For any 
for any 
are collinearly.
PP1. 
of 
belong to parabola 
with equation 
, 
and 
, 
are tangent to . Find 
, where 
is length of 
-median.Preliminary. We know that
for any point 
of parabola , where 
. For is well-known the vertex 
, the focus
, i.e. 
and the director 
with equation 
so that for any point 
there is the definition relation
. Thus, 
, i.e. 
.Proof 1 (analytic). Suppose w.l.o.g. that equation of parabola
is 
, where vertex is 
, focus is 
and equation of director is 
. Suppose that
. 
is intersection of tangents in 
, 
to parabola and which have equations 
. Thus, 
. Themidpoint of
![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
is 
and 
. Thus, 
and 
. The area![$S=[ABC]$](http://latex.artofproblemsolving.com/b/3/a/b3ae3d445111e4dd28be75922309d3270079368c.png)
is 
. With invariant transformations
get 
.Proof 2 (shorter, synthetic-analytic).
. Observe that 
and
, i.e. 
. Therefore, the area 
of is 
a.s.o.Propose four similar problems which use the remarkable relations
between the points 
and 
.[/size]AP1
Let be the parabola with equation , where 
. Consider a fixed point 
and two mobile points 
so that
. Find the locus of 
, where mean by 
- the tangent in the point 
to the parabola . Study some particular cases.AP2
Let be the parabola with equation , where . Consider a fixed circle 
, where 
and two mobile points so that
is tangent to 
. Find the locus of , where mean by - the tangent in the point to the parabola . Study someparticular cases. What is the locus of
if instead of the circle have the circle 
? What happen in the general case of some circle ?AP3
Let 
, 
be three parabolas with equations 
, 
where 
and 
. Consider two mobilepoints
so that the line 
is tangent to 
. Prove that the intersection of tangents to 
in the points 
, 
belongs to 
.AP4
Given are the parabola 
and the circle 
, where 
. The tangent in a mobile point 
which belongs to the circle cut theparabola in the points
, 
. Denote the intersection 
of the tangents in the points , to the parabola. Ascertain the geometrical locus of the point . PP2. Let 
be a function, where 
and 
. Let be a line so that 
.For any
the tangent to 
in the point 
cut again in the point 
. Prove that the points 
are collinearly. Proof. Let
where . Thus, 
![$\left(x_2-x_1\right)\left(x_3-x_1\right)\left(x_3-x_2\right)\left[m\left(x_1+x_2+x_3\right)+n\right]=0\iff$](http://latex.artofproblemsolving.com/1/6/a/16a0196103c91b80c6ee64ef400396181d6783d6.png)
. For any 
for any 
are collinearly.This post has been edited 63 times. Last edited by Virgil Nicula, Nov 27, 2015, 6:50 AM
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