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{"name": "Equation of tangent to Parametric curve", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Equation of tangent to Parametric curve", "tags": [], "metadata": {"description": "Find the equation of the tangent to a curve defined parametrically", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A curve is defined by the parametric equations $x=\\var{a}t+\\frac{\\var{b}}{t^2}, y = \\var{a}t-\\frac{\\var{b}}{t^2}$</p>", "advice": "<p>(i) To find $P$ we substitute $t=\\var{t}$ into&nbsp;$x=\\var{a}t+\\frac{\\var{b}}{t^2}, y = \\var{a}t-\\frac{\\var{b}}{t^2}$</p>\n<p>Hence $x=\\var{a}\\times\\var{t}+\\frac{\\var{b}}{\\var{t}^2} = \\var{a*t} + \\var{b/t^2}=\\var{x}$</p>\n<p>and $y=\\var{a}\\times\\var{t}-\\frac{\\var{b}}{\\var{t}^2} = \\var{a*t} - \\var{b/t^2}=\\var{y}$</p>\n<p>Hence $P = (\\var{x},\\var{y})$</p>\n<p></p>\n<p>(ii) To find the equation of the tangent, we need to find the gradient $m$, which is simply the value of $dy/dx$ at the point $P$. Since we are working with parametric equations, it is easiest to calculate $dy/dx=\\frac{dy}{dt} / \\frac{dx}{dt}$, and evaluate these when $t = \\var{t}$, corresponding to the point $P$</p>\n<p>Now&nbsp;$\\frac{dy}{dt} = \\var{a}-\\frac{\\var{b}}{t^3} =&nbsp;\\var{a}-\\frac{\\var{b}}{\\var{t}^3} =\\var{ydot}$ at the point $P$</p>\n<p>and $\\frac{dx}{dt} = \\var{a}+\\frac{\\var{b}}{t^3} =&nbsp;\\var{a}+\\frac{\\var{b}}{\\var{t}^3}= \\var{xdot}$ at the point $P$</p>\n<p>Hence $m=dy/dx = \\frac{\\var{ydot}}{\\var{xdot}}=\\simplify[all,fractionNumbers]{{m}}$</p>\n<p></p>\n<p>Now we can find $c$ by substituting $m=\\simplify[all,fractionNumbers]{{m}}, x=\\var{x}$ and $y=\\var{y}$ into $y = mx+c$</p>\n<p>Hence $c=y-mx=\\var{y}-\\simplify[all,fractionNumbers]{{m}}\\times \\var{x} = \\simplify[all,fractionNumbers]{{c}}$</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,4,5)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,5)", "description": "", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(2,4,5)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "ydot/xdot", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "y-m*x", "description": "", "templateType": "anything"}, "xdot": {"name": "xdot", "group": "Ungrouped variables", "definition": "a-2*b/t^3", "description": "", "templateType": "anything"}, "ydot": {"name": "ydot", "group": "Ungrouped variables", "definition": "a+2*b/t^3", "description": "", "templateType": "anything"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "a*t+b/t^2", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "a*t-b/t^2", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "xdot<>0", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "t", "m", "c", "xdot", "ydot", "x", "y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p></p>\n<p>(i) At the point $P$ on the curve, $t=\\var{t}$. Find the co-ordinates of $P$.</p>\n<p>$P$ = ([[0]],[[1]])</p>\n<p></p>\n<p>(ii) Find the equation of the tangenet to the curve at point $P$, in the form $y = mx+c$.</p>\n<p>$m$ =&nbsp;[[2]] (give your answer as a fraction)</p>\n<p>$c$ =&nbsp;[[3]]&nbsp;</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{x}", "maxValue": "{x}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{y}", "maxValue": "{y}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{m}", "maxValue": "{m}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{c}-0.001", "maxValue": "{c}+0.001", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}