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{"name": "Katherine's copy of Q2 Given a line, Coordinate Geometry ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Katherine's copy of Q2 Given a line, Coordinate Geometry ", "preamble": {"js": "", "css": ""}, "extensions": [], "statement": "<p>L is the line $\\var{l[0]}x + \\var{l[1]}y = \\var{l[2]}$</p>\n<p></p>", "rulesets": {}, "parts": [{"gaps": [{"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "{l[2]}/{l[0]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "all"}, {"notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "0", "marks": 1, "type": "numberentry", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "minValue": "0"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Find the point a, where L intersects the x-axis.</p>\n<p>Give answer in fraction form.</p>\n<p>a = ([[0]]&nbsp;,&nbsp;[[1]])</p>"}, {"gaps": [{"notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "0", "marks": 1, "type": "numberentry", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "minValue": "0"}, {"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "{l[2]}/{l[1]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Find the point b, where L intersects the y-axis.</p>\n<p>Give answer in fraction form.</p>\n<p>a = ([[0]]&nbsp;,&nbsp;[[1]])</p>"}, {"gaps": [{"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "{l[1]}/{l[0]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "ALL"}, {"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "{constant4a}/{l[0]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Find the equation of the line&nbsp;perpendicular to L and passing through c($\\var{px},\\var{py}$).</p>\n<p>Give answer in fraction form.</p>\n<p>y =&nbsp;[[0]]x + [[1]]</p>"}, {"gaps": [{"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "-{l[0]}/{l[1]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "all"}, {"vsetrange": [0, 1], "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "variableReplacements": [], "showCorrectAnswer": true, "answer": "{constanta}/{l[1]}", "marks": 1, "type": "jme", "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5, "answersimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Find the equation of the line parallel to L&nbsp;and passing through d($\\var{px1},\\var{py1}$).</p>\n<p>Give answer in fraction form.</p>\n<p>y =&nbsp;[[0]]x + [[1]]</p>"}, {"gaps": [{"notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "x5", "marks": 1, "type": "numberentry", "correctAnswerFraction": true, "mustBeReducedPC": 0, "allowFractions": true, "minValue": "x5"}, {"notationStyles": ["plain", "en", "si-en"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "y5", "marks": 1, "type": "numberentry", "correctAnswerFraction": true, "mustBeReducedPC": 0, "allowFractions": true, "minValue": "y5"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Find the mid-point of the line segment cd.</p>\n<p>Give answer to 2 decimal points.</p>\n<p>([[0]],[[1]])</p>"}, {"gaps": [{"precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "correctAnswerStyle": "plain", "strictPrecision": false, "mustBeReduced": false, "showPrecisionHint": false, "variableReplacements": [], "showCorrectAnswer": true, "maxValue": "{dis}", "marks": 1, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "mustBeReducedPC": 0, "allowFractions": false, "minValue": "{dis}", "precision": "2"}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "prompt": "<p>Calculate the distance from c to d.</p>\n<p>Give answer to 2 decimal places.</p>\n<p>ans = [[0]]</p>"}], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "<p>&nbsp;$\\var{l[0]}x + \\var{l[1]}y = \\var{l[2]}$</p>\n<p>a)&nbsp;L intersects the x axis when $y=0$</p>\n<p>$\\var{l[0]}x + \\var{l[1]}(0) = \\var{l[2]}$</p>\n<p>$\\var{l[0]}x = \\var{l[2]}$</p>\n<p>$x = \\frac{\\var{l[2]}}{\\var{l[0]}}$</p>\n<p>Point $= (\\frac{\\var{l[2]}}{\\var{l[0]}},0)$</p>\n<p>b) L intersects the&nbsp;y&nbsp;axis when $ x=0$</p>\n<p>&nbsp;$\\var{l[0]}0 + \\var{l[1]}y = \\var{l[2]}$</p>\n<p>&nbsp;$ \\var{l[1]}y = \\var{l[2]}$</p>\n<p>&nbsp;$ y = \\frac{\\var{l[2]}}{\\var{l[1]}}$</p>\n<p>Point $= (0,\\frac{\\var{l[2]}}{\\var{l[1]}})$&nbsp;</p>\n<p>c) To find the equation of a line we use the formula&nbsp;$y - y_1 = m(x - x_1)$, where $(x_1,y_1)$ is a&nbsp;point on the line and $m$ is the slope.</p>\n<p>We are given the point c($\\var{px},\\var{py}$). So, we need to find the slope $m$.</p>\n<p>Slope of L &nbsp;= $ -\\frac{\\var{l[0]}}{\\var{l[1]}}$. Since the line is perpendicular&nbsp;to L&nbsp;the product of the slopes is -1. i.e. $m\\times (-\\frac{\\var{l[0]}}{\\var{l[1]}})) = -1 $</p>\n<p>$m = &nbsp;\\frac{\\var{l[1]}}{\\var{l[0]}} $</p>\n<p>$Y - (\\var{py}) = \\frac{\\var{l[1]}}{\\var{l[0]}}(X - (\\var{px}))$</p>\n<p>Rearranging we get</p>\n<p>$Y=\\frac{{\\var{l[1]}}}{{\\var{l[0]}}}x+\\frac{\\var{constant4a}}{\\var{l[0]}}$</p>\n<p>d) Using the slope of L&nbsp;from&nbsp;part c and the formula $y - y_1 = m(x - x_1)$, where $(x_1,y_1)$ is the point d$(\\var{px1},\\var{py1})$.</p>\n<p>Since the line is parallel to L&nbsp;the slope is the same.</p>\n<p>slope of L(m)&nbsp;=&nbsp;$ -\\frac{\\var{l[0]}}{\\var{l[1]}}$</p>\n<p>$y - (\\var{py1})&nbsp;= &nbsp;-\\frac{\\var{l[0]}}{\\var{l[1]}}$(x - \\var{px1})$</p>\n<p>&nbsp;$y= -\\frac{\\var{l[0]}}{\\var{l[1]}}x + &nbsp;\\var{constanta}/\\var{l[1]}$</p>\n<p>e) Using the formula $(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2})$, where $(x_1,y_1) = c(\\var{px},\\var{py})$ and $(x_2,y_2) = d(\\var{px1},\\var{py1})$.</p>\n<p>$(\\frac{\\var{px}+\\var{px1}}{2},\\frac{\\var{py}+\\var{py1}}{2}) = (\\var{x5},\\var{y5})$</p>\n<p>f)&nbsp;Using the formula dis = $\\sqrt{(x_2-x_1)^2 + (y_2-y-1)^2}$,&nbsp;where $(x_1,y_1) = c(\\var{px},\\var{py})$ and $(x_2,y_2) = d(\\var{px1},\\var{py1})$.</p>\n<p>$\\sqrt{(\\var{px1}-\\var{px})^2 + (\\var{py1})-(\\var{py})))^2}={\\var{dis}}$</p>\n<p>Then round to 2 decimal places.</p>\n<p></p>", "variable_groups": [{"name": "iv", "variables": ["ya", "xa", "constanta", "gcd1a", "gcd2a", "gcda", "y1a", "x1a", "constant1a", "py1", "px1"]}, {"name": "iii", "variables": ["m", "py", "px", "y4a", "x4a", "constant4a", "gcd4a", "gcd4b", "gcd4", "x4", "y4", "constant4"]}, {"name": "v", "variables": ["x5", "y5"]}, {"name": "vi", "variables": ["dis"]}], "ungrouped_variables": ["l", "p_a", "p_b", "l_s", "y", "constant", "x", "gcd1", "gcd2", "gcd", "y1", "x1", "constant1", "neg", "pos"], "variables": {"p_b": {"name": "p_b", "group": "Ungrouped variables", 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"definition": "ya/gcda", "templateType": "anything"}, "gcd4b": {"name": "gcd4b", "group": "iii", "description": "", "definition": "gcd(constant4a,y4a)", "templateType": "anything"}, "ya": {"name": "ya", "group": "iv", "description": "", "definition": "l_s", "templateType": "anything"}, "px1": {"name": "px1", "group": "iv", "description": "", "definition": "random(2..8)", "templateType": "anything"}, "gcd4": {"name": "gcd4", "group": "iii", "description": "", "definition": "gcd(gcd4a,gcd4b)", "templateType": "anything"}, "xa": {"name": "xa", "group": "iv", "description": "", "definition": "l[0]", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "description": "", "definition": "(neg[2]-pos[0])", "templateType": "anything"}, "x4": {"name": "x4", "group": "iii", "description": "", "definition": "l[1]/l[0]", "templateType": "anything"}, "gcd2": {"name": "gcd2", "group": "Ungrouped variables", "description": "", "definition": "gcd(constant,y)", "templateType": 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"shuffle(-4..-1)[0..3]", "templateType": "anything"}, "dis": {"name": "dis", "group": "vi", "description": "", "definition": "sqrt(((px1-px)^2) + ((py1-py)^2))", "templateType": "anything"}, "p_a": {"name": "p_a", "group": "Ungrouped variables", "description": "", "definition": "l[2]/l[0]", "templateType": "anything"}, "py1": {"name": "py1", "group": "iv", "description": "", "definition": "random(-5..-2 except py)", "templateType": "anything"}, "m": {"name": "m", "group": "iii", "description": "", "definition": "-1/l_s", "templateType": "anything"}, "gcd": {"name": "gcd", "group": "Ungrouped variables", "description": "", "definition": "gcd(gcd1,gcd2)", "templateType": "anything"}, "l": {"name": "l", "group": "Ungrouped variables", "description": "", "definition": "shuffle(2..6)[0..3]", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "description": "", "definition": "(neg[1]-neg[0])", "templateType": "anything"}, "px": {"name": "px", "group": "iii", 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"description": "", "definition": "((neg[1]-neg[0])*-pos[0])-((neg[2]-pos[0])*-neg[0])", "templateType": "anything"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Practice finding parallel and perpendicular lines to a given line.</p>\n<p>rebelmaths</p>"}, "functions": {}, "type": "question", "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}]}], "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}