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{"name": "Rectangular Hyperbola 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d"], "name": "Rectangular Hyperbola 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Don't forget the following when trying to find asymptotes:</p>\n<ul>\n<li>The asyptote that is parallel to the $x$-axis is found by equating the coefficient of the highest order of $y$ to zero.</li>\n<li>Similarly, the asyptote that is parallel to the $y$-axis is found by equating the coefficient of the highest order of $x$ to zero.</li>\n<li>Transposition is almost always required!</li>\n</ul>\n<p>Steps:</p>\n<ol>\n<li>Transpose to eliminate the fraction: $y(\\var{c}x+\\var{d}) = \\var{a}x+\\var{b}$</li>\n<li>At this stage we have a coefficient for the highest order of $y$.</li>\n<ul>\n<li>Equate this to zero: $\\var{c}x+\\var{d}=0$</li>\n<li>Transpose and solve for $x$: &nbsp;$x=-\\var{d}/\\var{c}$</li>\n<li>This is where our asymptote parallel to the $y$-axis crosses the $x$-axis.</li>\n</ul>\n<li>Going back to step 1, multiply out the bracket: $\\var{c}xy+\\var{d}y =&nbsp;\\var{a}x+\\var{b}$</li>\n<li>Collect $x$ terms together: $\\var{c}xy-\\var{a}x+\\var{d}y-\\var{b} = 0$</li>\n<li>Factorise for $x$: $x(\\var{c}y-\\var{a})+\\var{d}y-\\var{b} = 0$</li>\n<li>Equate the coefficient of $x$ to zero:</li>\n<ul>\n<li>\\var{c}y-\\var{a}=0</li>\n<li>Transpose and solve for $y$: $y=\\var{a}/\\var{c}$</li>\n<li>This is where our asymptote parallel to the $x$-axis crosses the $y$-axis.</li>\n</ul>\n</ol>", "rulesets": {}, "parts": [{"prompt": "<p>Where does the asymptote that is parallel to the $x-axis$ intersect the $y-axis$?</p>\n<p>&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{a}/{c}", "minValue": "{a}/{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Where does the asymptote that is parallel to the $y-axis$ intersect the $x-axis$?</p>\n<p>&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{-d}/{c}", "minValue": "{-d}/{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>You are given the equation \\[\\simplify[all]{y({c}x+{d})=({a}x+{b})}.\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "a+c=0"}, "variables": {"a": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-12..12 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "<p>In this question students are asked to find the asymptotes that are parallel to the $x-$ and $y-$axis.</p>", "licence": "All rights reserved"}, "type": "question", "contributors": [{"name": "Jon Stammers", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/891/"}]}]}], "contributors": [{"name": "Jon Stammers", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/891/"}]}