// Numbas version: exam_results_page_options
{"name": "Determinar el centro y radio de una circunferencia completando cuadrados", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Determinar el centro y radio de una circunferencia completando cuadrados", "tags": [], "metadata": {"description": "<p>Completa el cuadrado dos veces para determinar el radio y el centro de un c&iacute;rculo.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<div class=\"tlid-input input\">\n<div class=\"source-wrap\">\n<div class=\"input-full-height-wrapper tlid-input-full-height-wrapper\">\n<div class=\"source-input\">\n<div class=\"source-header\">\n<div class=\"go-wrap\">Considere la ecuaci&oacute;n general de la circunferencia \\[\\simplify[basic,fractionNumbers]{{scoeff}x^2+{lcoeff}x+{scoeff}y^2+{m}y+{k}={c}}.\\]</div>\n<div class=\"go-wrap\">Reducir la ecuaci&oacute;n a la forma estandar y responde:</div>\n</div>\n</div>\n</div>\n</div>\n</div>", "advice": "<ol>\n<li>Recuerde que&nbsp;</li>\n<li style=\"text-align: center;\">$(x+a)^2=x^2+2ax+a^2$</li>\n<li>es un cuadrado perfecto.&nbsp; Ahora, note que si $b=2a$ esta ecuaci&oacute;n llegar&iacute;a a ser</li>\n<li style=\"text-align: center;\">$\\left(x+\\frac{b}{2}\\right)^2=x^2+bx+\\left(\\frac{b}{2}\\right)^2$.</li>\n<li style=\"text-align: left;\">Completamos los cuadrados:</li>\n<li></li>\n</ol>\n<table>\n<tbody>\n<tr style=\"height: 34px;\">\n<td style=\"height: 34px;\">$\\simplify[basic,basic,fractionNumbers]{{scoeff}x^2+{lcoeff}x+{scoeff}y^2+{m}y+{k}}$</td>\n<td style=\"text-align: center; height: 34px;\">$=$</td>\n<td style=\"text-align: left; height: 34px;\">$\\var{c}$</td>\n<td style=\"text-align: left; height: 34px;\"></td>\n</tr>\n<tr style=\"height: 85px;\">\n<td style=\"text-align: right; height: 85px;\">$\\simplify[basic,basic,fractionNumbers]{{scoeff}x^2+{lcoeff}x+{scoeff}y^2+{m}y}$&nbsp;</td>\n<td style=\"text-align: center; height: 85px;\">$=$</td>\n<td style=\"text-align: left; height: 85px;\">$\\simplify[basic,fractionNumbers]{{c-k}}$</td>\n<td style=\"text-align: left; height: 85px;\">(obtener todas las constantes en el lado derecho)</td>\n</tr>\n<tr style=\"height: 119px;\">\n<td style=\"text-align: right; height: 119px;\">$\\simplify[basic,fractionNumbers]{x^2+{lcoeff}/{scoeff}x}+\\simplify[basic,fractionNumbers]{y^2+{m}/{scoeff}y}$</td>\n<td style=\"text-align: center; height: 119px;\">$=$</td>\n<td style=\"text-align: left; height: 119px;\">$\\simplify[basic,fractionNumbers]{{(c-k)/scoeff}}$</td>\n<td style=\"text-align: left; height: 119px;\">\n<div class=\"tlid-input input\">\n<div class=\"source-wrap\">\n<div class=\"input-full-height-wrapper tlid-input-full-height-wrapper\">\n<div class=\"source-input\">\n<div class=\"source-header\">\n<div class=\"go-wrap\"></div>\n</div>\n<div class=\"source-footer-wrap source-or-target-footer\">\n<div class=\"source-input-tools\" id=\"gt-input-tool\">\n<div id=\"itamenu\">\n<div class=\"goog-container goog-container-vertical\" tabindex=\"-1\">(Divide cada t&eacute;rmino por el coeficiente de $x^2 $)</div>\n</div>\n</div>\n</div>\n</div>\n</div>\n</div>\n</div>\n</td>\n</tr>\n<tr style=\"height: 85px;\">\n<td style=\"text-align: right; height: 85px;\">$\\simplify[basic,fractionNumbers]{x^2+{lcoeff/scoeff}x+{lcoeff^2/(4*scoeff^2)}}+\\simplify[basic,fractionNumbers]{y^2+{m/scoeff}y+{m^2/(4*scoeff^2)}}$</td>\n<td style=\"text-align: center; height: 85px;\">$=$</td>\n<td style=\"text-align: left; height: 85px;\">$\\simplify[basic,fractionNumbers]{{(c-k)/scoeff}}+\\simplify{{lcoeff^2}/{4*scoeff^2}}+\\simplify{{m^2}/{4*scoeff^2}}$</td>\n<td style=\"text-align: left; height: 85px;\">\n<p>(para $ x $ y $ y $: agregue a ambos lados el cuadrado de la mitad del coeficiente)</p>\n</td>\n</tr>\n<tr style=\"height: 102px;\">\n<td style=\"text-align: right; height: 102px;\">$(\\simplify[basic,fractionNumbers]{x+{-xcentre}})^2+(\\simplify[basic,fractionNumbers]{y+{-ycentre}})^2$</td>\n<td style=\"text-align: center; height: 102px;\">$=$</td>\n<td style=\"text-align: left; height: 102px;\">$\\simplify[basic,fractionNumbers]{{radius^2}}$</td>\n<td style=\"text-align: left; height: 102px;\">(reescribe el lado izquierdo como dos cuadrados perfectos)</td>\n</tr>\n<tr style=\"height: 34px;\">\n<td style=\"text-align: right; height: 34px;\">$(\\simplify[basic,fractionNumbers]{x+{-xcentre}})^2+(\\simplify[basic,fractionNumbers]{y+{-ycentre}})^2$</td>\n<td style=\"text-align: center; height: 34px;\">$=$</td>\n<td style=\"text-align: left; height: 34px;\">$\\left(\\simplify[basic,fractionNumbers]{{radius}}\\right)^2$</td>\n<td style=\"text-align: left; height: 34px;\"></td>\n</tr>\n</tbody>\n</table>\n<p><br/>De esta ecuaci&oacute;n podemos concluir que la ecuaci&oacute;n es de un c&iacute;rculo de radio $\\simplify[basic,fractionNumbers]{{radius}}$ with&nbsp;centre $\\left(\\simplify[basic,fractionNumbers]{{xcentre}},\\simplify[basic,fractionNumbers]{{ycentre}}\\right)$.</p>", "rulesets": {}, "extensions": [], "variables": {"sqrtargtop": {"name": "sqrtargtop", "group": "Ungrouped variables", "definition": "sqrt(argtop)", "description": "", "templateType": "anything"}, "argtop": {"name": "argtop", "group": "Ungrouped variables", "definition": "lcoeff^2-4*ccoeff*scoeff", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "argbot": {"name": "argbot", "group": "Ungrouped variables", "definition": "4*scoeff^2", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "if(switch=0,ccoeff+c+m^2/(4*a*b),m^2/(4a*b)+c)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-1,1)*random(2,a,b,a*b,2*a,2*b,2*a*b)", "description": "<p>coeff of y</p>", "templateType": "anything"}, "sqrtargbot": {"name": "sqrtargbot", "group": "Ungrouped variables", "definition": "sqrt(argbot)", "description": "", "templateType": "anything"}, "lengthdet": {"name": "lengthdet", "group": "Ungrouped variables", "definition": "abs(a*d-b*c)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "radius": {"name": "radius", "group": "Ungrouped variables", "definition": "if(switch=0,abs((a*d-b*c)/(2a*b)),abs((a*d+b*c)/(2a*b)))", "description": "", "templateType": "anything"}, "ycentre": {"name": "ycentre", "group": "Ungrouped variables", "definition": "-m/(2*scoeff)", "description": "", "templateType": "anything"}, "div": {"name": "div", "group": "Ungrouped variables", "definition": "lcoeff/scoeff", "description": "", "templateType": "anything"}, "dd": {"name": "dd", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "xcentre": {"name": "xcentre", "group": "Ungrouped variables", "definition": "-lcoeff/(2*scoeff)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "lcoeff": {"name": "lcoeff", "group": "Ungrouped variables", "definition": "a*d+b*c", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(c*b=a*dd,dd+1,dd)", "description": "", "templateType": "anything"}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "(lcoeff^2-4*scoeff*ccoeff)", "description": "", "templateType": "anything"}, "scoeff": {"name": "scoeff", "group": "Ungrouped variables", "definition": "a*b", "description": "", "templateType": "anything"}, "switch": {"name": "switch", "group": "Ungrouped variables", "definition": "random(0,0,0,1)", "description": "<p>if 0 then small radius if 1 then larger (on average)</p>", "templateType": "anything"}, "ccoeff": {"name": "ccoeff", "group": "Ungrouped variables", "definition": "c*d", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "dd", "d", "scoeff", "lcoeff", "ccoeff", "disc", "lengthdet", "div", "argtop", "argbot", "sqrtargtop", "sqrtargbot", "m", "k", "xcentre", "ycentre", "radius", "switch"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>La ecuaci&oacute;n representa una circunferencia c&iacute;rculo con radio. [[0]]&nbsp; y centro en el punto $\\large($ [[1]],&nbsp;[[2]]&nbsp;$\\large)$.</p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Escriba la ecuaci&oacute;n del c&iacute;rculo en forma est&aacute;ndar, $(x-a)^2 + (y-b)^2 = r^2$, completando el cuadrado para los t&eacute;rminos $ x $ y tambi&eacute;n para los t&eacute;rminos $y$. Una vez hecho esto, debe reconocer la ecuaci&oacute;n de un c&iacute;rculo de radio $r$ con el centro $(a,b)$.</p>\n<p></p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{radius}", "maxValue": "{radius}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{xcentre}", "maxValue": "{xcentre}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ycentre}", "maxValue": "{ycentre}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}