## TPTP Problem File: RAL060^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL060^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Algebraic curves)
% Problem  : Osaka University, 2003, Science Course, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : (1) Consider an ellipse on a plane that has principal axes
%            (major and minor axes) parallel to the coordinate axes and is in
%            contact with both the x and y axes. Let a be the x coordinate of
%            the center of the ellipse. Find the range of the value of a such
%            that there exists such an ellipse that passes through the point
%            A(1, 2), where circles are regarded as special cases of ellipses.
%            (2) For a that gives just two ellipses satisfying the description
%            in (1), let B and C be the centers of the 2 ellipses and let the
%            function S(a) be the area of triangle ABC. Then, draw the graph of
%            S(a).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Osaka-2003-Ri-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45334 (2210 equality;22711 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39627 ( 104   ~; 233   |;1176   &;35988   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1200   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8063 (  66 sgn;7085   !; 434   ?; 408   ^)
%                                         (8063   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1980 (   6 prd;   9 fun;  23 num;1942 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-05-14
%            ^ [V_a_dot_0: \$real] :
%              ( \$greatereq @ V_a_dot_0 @ ( \$quotient @ 1.0 @ 2.0 ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

thf(p1_qustion,conjecture,
( 'find/1' @ \$real
@ ( ^ [V_a: \$real] :
? [V_A: '2d.Point',V_C: '2d.Shape',V_b: \$real,V_s: \$real,V_t: \$real] :
( ( V_A
= ( '2d.point/2' @ 1.0 @ 2.0 ) )
& ( V_C
= ( '2d.graph-of-implicit-function/1'
@ ( ^ [V_x: \$real,V_y: \$real] :
( \$sum @ ( \$quotient @ ( '^/2' @ ( \$difference @ V_x @ V_a ) @ 2.0 ) @ ( '^/2' @ V_s @ 2.0 ) ) @ ( \$sum @ ( \$quotient @ ( '^/2' @ ( \$difference @ V_y @ V_b ) @ 2.0 ) @ ( '^/2' @ V_t @ 2.0 ) ) @ -1.0 ) ) ) ) )
& ( '2d.tangent/2' @ V_C @ '2d.x-axis/0' )
& ( '2d.tangent/2' @ V_C @ '2d.y-axis/0' )
& ( '2d.on/2' @ V_A @ V_C ) ) ) )).
%------------------------------------------------------------------------------
```