// Numbas version: exam_results_page_options
{"name": "Luis's copy of Intersection of two circles in two points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["checked2015", "equation of a circle", "equation of a straight line", "Equation of a straight line", "intersection of two circles", "quadratic equation"], "extensions": [], "statement": "\n    <p>Find the points where the circles:</p>\n  <ul>\n  <li>$\\simplify[std]{x^2+y^2 -{2*k2}x -{2*l2}y+{2*k2*a+2*l2*b-a^2-b^2}}=0$\n  </li>\n  </ul>\n  <p>&nbsp;</p>\n  <ul>\n  <li>and $\\simplify[std]{x^2+y^2 -{2*k}x -{2*l}y+{2*k*a+2*l*b-a^2-b^2}}=0$</li>\n  </ul>\n  <p>&nbsp;</p>\n  meet.\n  <p>&nbsp;</p>\n  \n  <p>Enter the coordinates for the two points, starting with the point with the smaller $x$-coordinate.</p>\n  <p>Note that, for example, $-3$ is smaller than $-1$. Thus if the points of intersection were $(-1, 4)$ and $(-3,6)$ you would enter $(-3,6)$ for the first point and $(-1, 4)$ for the second.</p>\n  <p>Input the coordinates as integers.</p>\n  \n  \n    ", "name": "Luis's copy of Intersection of two circles in two points", "variable_groups": [], "ungrouped_variables": ["a", "k2", "c", "b", "d", "g", "f", "t2", "s1", "m", "l", "n", "q", "p", "l2", "b1", "k", "t"], "parts": [{"unitTests": [], "gaps": [{"unitTests": [], "vsetRangePoints": 5, "customMarkingAlgorithm": "", "variableReplacements": [], "showPreview": true, "answer": "{m}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "vsetRange": [0, 1], "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "expectedVariableNames": [], "answerSimplification": "std", "checkVariableNames": false, "notallowed": {"partialCredit": 0, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "showStrings": false, "strings": ["."]}, "marks": 1}, {"unitTests": [], "vsetRangePoints": 5, "customMarkingAlgorithm": "", "variableReplacements": [], "showPreview": true, "answer": "{n}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "vsetRange": [0, 1], "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "expectedVariableNames": [], "answerSimplification": "std", "checkVariableNames": false, "notallowed": {"partialCredit": 0, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "showStrings": false, "strings": ["."]}, "marks": 1}, {"unitTests": [], "vsetRangePoints": 5, "customMarkingAlgorithm": "", "variableReplacements": [], "showPreview": true, "answer": "{p}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "vsetRange": [0, 1], "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "expectedVariableNames": [], "answerSimplification": "std", "checkVariableNames": false, "notallowed": {"partialCredit": 0, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "showStrings": false, "strings": ["."]}, "marks": 1}, {"unitTests": [], "vsetRangePoints": 5, "customMarkingAlgorithm": "", "variableReplacements": [], "showPreview": true, "answer": "{q}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showFeedbackIcon": true, "vsetRange": [0, 1], "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "expectedVariableNames": [], "answerSimplification": "std", "checkVariableNames": false, "notallowed": {"partialCredit": 0, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "showStrings": false, "strings": ["."]}, "marks": 1}], "customMarkingAlgorithm": "", "variableReplacements": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "type": "gapfill", "marks": 0, "scripts": {}, "prompt": "<p>([[0]],[[1]]) and ([[2]],[[3]])</p>"}], "variables": {"l2": {"definition": "(b+d+t2*a-t2*c)/2", "name": "l2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s1": {"definition": "random(-1,1)", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "l": {"definition": "(b+d+t*a-t*c)/2", "name": "l", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "p": {"definition": "if(a<c,c,a)", "name": "p", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "m": {"definition": "if(a<c,a,c)", "name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "g": {"definition": "(b*c-a*d)/(c-a)", "name": "g", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "t2": {"definition": "random(-5..5 except t)", "name": "t2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "n": {"definition": "if(a<c,b,d)", "name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "q": {"definition": "if(a<c,d,b)", "name": "q", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c": {"definition": "a+Random(2,4)*s1", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "k": {"definition": "(a+c+t*d-t*b)/2", "name": "k", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "t": {"definition": "random(-5..5 except 0)", "name": "t", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "f": {"definition": "(b-d)/(a-c)", "name": "f", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "d": {"definition": "random(-5..5)", "name": "d", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"definition": "b1+d", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b1": {"definition": "random(-2,2,-4,4)", "name": "b1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"definition": "random(1,-1)*random(1..4)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "k2": {"definition": "(a+c+t2*d-t2*b)/2", "name": "k2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the points of intersection of two circles.</p>"}, "preamble": {"css": "", "js": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "advice": "\n    <p>Subtract the equation for the first circle from the equation for the second. You get $\\simplify[std]{{2*k2-2*k}x+ {2*l2-2*l}y+{2*k*a+2*l*b-2*k2*a-2*l2*b}}=0$.</p>\n    <p>This can be rearranged as: $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$</p>\n    <p>Substitute for $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ in the equation of the circle.</p>\n  <p>We get $\\simplify[std]{x^2+(({b-d}/{a-c})x+{b*c-a*d}/{c-a})^2 -{2*k}x -{2*l}(({b-d}/{a-c})x+{b*c-a*d}/{c-a})+{2*k*a+2*l*b-a^2-b^2}}=0$.</p>\n  <p>Collecting terms we get $\\simplify[std]{{1+ (b-d)^2/(a-c)^2} x^2+{(-2)*(b-d)(b*c-a*d)/(a-c)^2-2*k  -(2*l)*(b-d)/(a-c)}x+{(b*c-a*d)^2/(c-a)^2 -2*l*(b*c-a*d)/(c-a)+2*k*a+2*l*b-a^2-b^2}}=0$.</p>\n  <p>Solving this quadratic equation for $x$ gives solutions $x=\\var{a}$ and $x=\\var{c}$.</p>\n  <p>Substitute these values of $x$ into the equation $\\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ to obtain the $y$-coordinates: $y=\\var{b}$ (when $x=\\var{a}$) and $y=\\var{d}$ (when $x=\\var{c}$).</p>\n  <p>Enter the coordinate pairs in the order $(\\var{m},\\var{n})$, $(\\var{p}, \\var{q})$. </p>\n    ", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}