// Numbas version: exam_results_page_options
{"name": "Intersection of a straight line and a circle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)", "description": "", "name": "s1"}, "q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a<c,d,b)", "description": "", "name": "q"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1+d", "description": "", "name": "b"}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a<c,a,c)", "description": "", "name": "m"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(b-d)/(a-c)", "description": "", "name": "f"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2,2,-4,4)", "description": "", "name": "b1"}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5 except 0)", "description": "", "name": "t"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)*random(1..4)", "description": "", "name": "a"}, "k": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(a+c+t*d-t*b)/2", "description": "", "name": "k"}, "l": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(b+d+t*a-t*c)/2", "description": "", "name": "l"}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a<c,b,d)", "description": "", "name": "n"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "d"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(b*c-a*d)/(c-a)", "description": "", "name": "g"}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a<c,c,a)", "description": "", "name": "p"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+Random(2,4)*s1", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s1", "m", "l", "n", "q", "p", "b1", "k", "t"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "name": "Intersection of a straight line and a circle", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "{m}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{n}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{p}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{q}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers as appropriate and not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>([[0]],[[1]]) and ([[2]],[[3]])</p>", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n  <p>Find the points where:</p>\n    <ul>\n    <li>the straight line $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$</li>\n    </ul>\n    <p>&nbsp;</p>\n    <ul>\n    <li>and the circle $\\simplify[std]{x^2+y^2 -{2*k}x -{2*l}y+{2*k*a+2*l*b-a^2-b^2}}=0$</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  ", "tags": ["SFY0001", "checked2015", "equation of a circle", "equation of a straight line", "quadratic equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n    \t\t\n    \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the points of intersection of a straight line and a circle.</p>"}, "advice": "\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  ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}