## DESCRIPTION
##   Testing for Open Sets
## ENDDESCRIPTION

## KEYWORDS('Open Set', 'Ball')
## 

## DBsubject('Calculus')
## DBchapter('')
## DBsection('')
## Date('9/5/2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')
           
DOCUMENT();	# This should be the first executable line in the problem.

loadMacros("PG.pl",      
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",      
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "MathObjects.pl",   
          );      
              
TEXT(beginproblem());
$showPartialCorrectAnswers = 1;
Context("Vector");

$a1 = non_zero_random(1, 10, 1);
$b1 = non_zero_random(1, 10, 1);
$r = random(1,4,1);
$rr = $r**2;
$a2 = non_zero_random(-5, 5, 1);
$b2 = non_zero_random(-5, 5, 1);
$a3 = random(-10, 10, 1);
$b3 = random($a3+1, $a3+10, 1);


$ans1 = min($a1,$b1);
$ans2 = abs(sqrt($a2**2+$b2**2)-$r);
$ans3 = ($b3-$a3)/sqrt(2);

@points = ("$a1, $b1", "$a2, $b2", "$a3, $b3");

@sets = ( "xy \gt 0", "x^2+y^2 \ne $rr", "y \gt x" );

@answers = ("$ans1", "$ans2", "$ans3");

@choice=NchooseK(3,2);
@subpoints=@points[@choice];
@subsets=@sets[@choice];
@subanswers=@answers[@choice];

$P0 = ColumnVector("<@subpoints[0]>");
$P1 = ColumnVector("<$subpoints[1]>");

Context()->texStrings;
BEGIN_TEXT
Given the point \( \mathbf a = $P0 \) in the set
\( S = \left\{"\{"\} \left(\begin{array}{c}x\\y\end{array}\right): $subsets[0] \right\}\), give the radius \( \delta \) of the largest ball \(B(\mathbf a,\delta)\) that is contained in \( S\).$BR
\{ans_rule(10)\} $BR

Given the point \( \mathbf a = $P1 \) in the set
\( S = \left\{"\{"\} \left(\begin{array}{c}x\\y\end{array}\right):  $subsets[1] \right\}\), give the radius \(\delta\) of the largest ball \(B(\mathbf a,\delta)\) that is contained in \( S\).$BR
\{ans_rule(10)\} $BR

END_TEXT

ANS(num_cmp([@subanswers]));
ENDDOCUMENT();	# This should be the last executable line in the problem.