// Numbas version: finer_feedback_settings
{"name": "2.6.1 Task 7", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/HELM2_fig26.png", "/srv/numbas/media/question-resources/HELM2_fig26.png"], ["question-resources/HELM2_fig27.png", "/srv/numbas/media/question-resources/HELM2_fig27.png"], ["question-resources/HELM2_6_Task3.PNG", "/srv/numbas/media/question-resources/HELM2_6_Task3.PNG"], ["question-resources/HELM2_6_Task3a.png", "/srv/numbas/media/question-resources/HELM2_6_Task3a.png"], ["question-resources/HELM2_6_Task4.png", "/srv/numbas/media/question-resources/HELM2_6_Task4.png"], ["question-resources/HELM2_fig29.png", "/srv/numbas/media/question-resources/HELM2_fig29.png"], ["question-resources/HELM2_6_Task6.png", "/srv/numbas/media/question-resources/HELM2_6_Task6.png"], ["question-resources/HELM2_fig30.png", "/srv/numbas/media/question-resources/HELM2_fig30.png"], ["question-resources/HELM2_fig28.png", "/srv/numbas/media/question-resources/HELM2_fig28.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2.6.1 Task 7", "tags": [], "metadata": {"description": "<p>Rearrange a homogeneous circle equation into the form (x-xc)^2+(y-yc)^2=r^2 to find the centre and radius.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If</p>\n<p style=\"padding-left: 40px;\">$\\var{eqn}$</p>\n<p>obtain the centre and radius of the circle that this equation represents.</p>", "advice": "<p>The original equation $\\var{eqn}$ becomes</p>\n<p style=\"padding-left: 40px;\">$(\\var{simplify(expression(\"x^2-2*x*\"+Cx),\"all\")}+\\var{Cx^2})-\\var{Cx^2} + (\\var{simplify(expression(\"y^2-2*y*\"+Cy),\"all\")}+\\var{Cy^2})-\\var{Cy^2}$&nbsp; <span data-jme-visible=\"const&gt;0\">$+$</span> <span data-jme-visible=\"const&lt;&gt;0\">$\\var{const}$</span> $=0$</p>\n<p style=\"padding-left: 40px;\">$\\var{simplify(expression(\"(x-\"+Cx+\")^2\"),\"all\")}+\\var{simplify(expression(\"(y-\"+Cy+\")^2\"),\"all\")}-\\var{Cx^2}-\\var{Cy^2}$ <span data-jme-visible=\"const&gt;0\">$+$</span> <span data-jme-visible=\"const&lt;&gt;0\">$\\var{const}$</span> $=0$</p>\n<p style=\"padding-left: 40px;\">$\\therefore\\quad \\var{simplify(expression(\"(x-\"+Cx+\")^2\"),\"all\")}+\\var{simplify(expression(\"(y-\"+Cy+\")^2\"),\"all\")}=\\var{R2}$</p>\n<p>which represents a circle with centre $(\\var{Cx},\\var{Cy})$ and radius $\\var{R}$.</p>\n<p>{graph}</p>", "rulesets": {}, "extensions": ["jsxgraph"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Cx": {"name": "Cx", "group": "The circle", "definition": "random(-5..5)", "description": "<p>Centre x value</p>", "templateType": "anything", "can_override": false}, "Cy": {"name": "Cy", "group": "The circle", "definition": "random(-5..5)", "description": "<p>The centre y-value</p>", "templateType": "anything", "can_override": false}, "R2": {"name": "R2", "group": "The circle", "definition": "random(1..10)", "description": "<p>The square of the radius</p>", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "The circle", "definition": "simplify(expression(\"sqrt(\"+R2+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "eqn": {"name": "eqn", "group": "The circle", "definition": "simplify(expression(\"x^2+y^2-2*x*\"+Cx+\"-2*y*\"+Cy+\"+\"+const+\"=0\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "const": {"name": "const", "group": "The circle", "definition": "eval(expression(Cx+\"*\"+Cx+\"+\"+Cy+\"*\"+Cy+\"-\"+R2))", "description": "", "templateType": "anything", "can_override": false}, "graph": {"name": "graph", "group": "the graph", "definition": "jessiecode(\n  400,400,[{xmin},{ymax},{xmax},{ymin}],\"r={R};Cx={Cx};Cy={Cy};\"+safe(\n  \"\"\"\n    X = point(Cx,-r+Cy) <<visible: false>>;\n    Y = point(r+Cx,Cy) <<visible: false>>;\n    Z = point(-r+Cx,Cy) <<visible: false>>;\n    circle(X,Y,Z) << strokeOpacity: 0.3 >>;\n    C=point(Cx,Cy) <<color:'blue', name:''>>;\n  \"\"\"),\n  [\n    \"axis\": true\n  ]\n)", "description": "", "templateType": "anything", "can_override": false}, "xmin": {"name": "xmin", "group": "the graph", "definition": "min(-1,Cx-ceil(eval(R))-1)", "description": "", "templateType": "anything", "can_override": false}, "ymin": {"name": "ymin", "group": "the graph", "definition": "min(-1,Cy-ceil(eval(R))-1)", "description": "", "templateType": "anything", "can_override": false}, "xmaxtemp": {"name": "xmaxtemp", "group": "the graph", "definition": "max(1,Cx+ceil(eval(R))+1)", "description": "", "templateType": "anything", "can_override": false}, "xmax": {"name": "xmax", "group": "the graph", "definition": "if(xmaxtemp-xmin > ymaxtemp-ymin,xmaxtemp,xmin+ymaxtemp-ymin)", "description": "", "templateType": "anything", "can_override": false}, "ymaxtemp": {"name": "ymaxtemp", "group": "the graph", "definition": "max(1,Cy+ceil(eval(R))+1)", "description": "", "templateType": "anything", "can_override": false}, "ymax": {"name": "ymax", "group": "the graph", "definition": "if(ymaxtemp-ymin > xmaxtemp-xmin,ymaxtemp,ymin+xmaxtemp-xmin)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "The circle", "variables": ["Cx", "Cy", "R2", "R", "const", "eqn"]}, {"name": "the graph", "variables": ["graph", "xmin", "ymin", "xmaxtemp", "ymaxtemp", "xmax", "ymax"]}], "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>Begin by completing the square separately on the $x-$terms and the $y-$terms.</p>\n<p>$\\var{simplify(expression(\"x^2-2*\"+Cx+\"*x\"),\"all\")} = (x + $[[0]]$)^2$ - [[1]]</p>\n<p>$\\var{simplify(expression(\"y^2-2*\"+Cy+\"*y\"),\"all\")} = (y + $[[2]]$)^2$ - [[3]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x_0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-Cx", "maxValue": "-Cx", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "x_c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Cx*Cx", "maxValue": "Cx*Cx", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "y_0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-Cy", "maxValue": "-Cy", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "y_c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Cy*Cy", "maxValue": "Cy*Cy", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>Now complete the problem:</p>\n<p>centre = ([[0]],[[1]])</p>\n<p>radius = [[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "centre x", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Cx", "maxValue": "Cx", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "centre y", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "Cy", "maxValue": "Cy", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": true, "customName": "radius", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{R}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.1", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}