## TPTP Problem File: RAL052^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL052^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Algebraic curves)
% Problem  : Hokkaido University, 2007, Science Course, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Consider the ellipse C_1 :x^2/{alpha}^2 +y^2/{beta}^2 = 1 and
%            hyperbola C_2 :x^2/a^2 -y^2/b^2 = 1. If C_1 and C_2 have the same
%            focuses, prove that the tangents of C_1 and C_2 intersect with
%            each other perpendicularly at the intersection of the two.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Hokkaido-2007-Ri-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45380 (2215 equality;22736 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39668 ( 109   ~; 233   |;1187   &;36012   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8070 (  66 sgn;7096   !; 429   ?; 409   ^)
%                                         (8070   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1982 (   6 prd;   9 fun;  23 num;1944 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Ukyo Suzuki; Generated: 2014-09-27
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1,conjecture,(
! [V_alpha: \$real,V_beta: \$real,V_a: \$real,V_b: \$real,V_C1: '2d.Shape',V_C2: '2d.Shape',V_p1: '2d.Point',V_p2: '2d.Point',V_p: '2d.Point',V_l: '2d.Shape',V_k: '2d.Shape'] :
( ( ( V_a != 0.0 )
& ( V_b != 0.0 )
& ( V_alpha != 0.0 )
& ( V_beta != 0.0 )
& ( V_C1
= ( '2d.graph-of-implicit-function/1'
@ ^ [V_x_dot_0: \$real,V_y_dot_0: \$real] :
( \$difference @ ( \$sum @ ( \$quotient @ ( '^/2' @ V_x_dot_0 @ 2.0 ) @ ( '^/2' @ V_alpha @ 2.0 ) ) @ ( \$quotient @ ( '^/2' @ V_y_dot_0 @ 2.0 ) @ ( '^/2' @ V_beta @ 2.0 ) ) ) @ 1.0 ) ) )
& ( V_C2
= ( '2d.graph-of-implicit-function/1'
@ ^ [V_x: \$real,V_y: \$real] :
( \$difference @ ( \$difference @ ( \$quotient @ ( '^/2' @ V_x @ 2.0 ) @ ( '^/2' @ V_a @ 2.0 ) ) @ ( \$quotient @ ( '^/2' @ V_y @ 2.0 ) @ ( '^/2' @ V_b @ 2.0 ) ) ) @ 1.0 ) ) )
& ( V_p1 != V_p2 )
& ( '2d.is-focus-of/2' @ V_p1 @ V_C1 )
& ( '2d.is-focus-of/2' @ V_p1 @ V_C2 )
& ( '2d.is-focus-of/2' @ V_p2 @ V_C1 )
& ( '2d.is-focus-of/2' @ V_p2 @ V_C2 )
& ( '2d.intersect/3' @ V_C1 @ V_C2 @ V_p )
& ( '2d.line-type/1' @ V_l )
& ( '2d.tangent/3' @ V_C1 @ V_l @ V_p )
& ( '2d.line-type/1' @ V_k )
& ( '2d.tangent/3' @ V_C2 @ V_k @ V_p ) )
=> ( '2d.perpendicular/2' @ V_l @ V_k ) ) )).

%------------------------------------------------------------------------------
```