Exercises for HELM Book 2.6.2.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], []], "questions": [{"name": "2.6.3 Circle radius and centre", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given a circle equation in the form (x-a)^2+(y-b)^2=r^2, write down the radius and centre coordinates.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write down the radius and the coordinates of the centre of the following circle
\n$\\var{equation}$
", "advice": "The centre and radius can be read directly from the equation:
\nCentre = ($\\var{centrex},\\var{centrey}$)
\nRadius = $\\var{radius}$
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\nRadius = [[2]]
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Obtain the equation of the circle with centre $C (\\var{centrex},\\var{centrey})$ and radius $\\var{radius}$.
", "advice": "The circle equation is given by $(x-x\\_centre)^2+(y-y\\_centre)^2 = radius^2$
\nSo the equation is $\\var{equation}$.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"centrex": {"name": "centrex", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "centrey": {"name": "centrey", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "radius": {"name": "radius", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "equation": {"name": "equation", "group": "Ungrouped variables", "definition": "simplify(expression(\"(x-\" + centrex + \")^2+(y-\"+ centrey + \")^2 = \" + radius*radius ),\"all\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["centrex", "centrey", "radius", "equation"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{equation}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "2.6.5 Circle radius and centre no 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Determine the radius and centre coordinates of a circle with equation expressed in expanded form.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write down the radius and the coordinates of the centre of the following circle
\n$\\var{equation}$
", "advice": "We need to begin by completing the square on the $x-$terms and on the $y-$ terms:
\n$\\var{equation}$
\n$\\var{completedsquares} = 0$
\n$\\var{simplifiedequation}$
So the circle has radius = $\\var{radius}$ and centre $C = (\\var{centrex},\\var{centrey})$
Centre = ( [[0]],[[1]] )
\nRadius = [[2]]
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Describe this region in words:
\n$\\var{eqn1}\\;$ and $\\;\\var{eqn2}$
", "advice": "This is the region outside the circle centred at the origin with radius $\\var{radius1}$.
\nThis is the region inside the circle centred at the origin with radius $\\var{radius2}$. This region is referred to as a disc.
\nThis is the annular ring between the circles centred at the origin with radii $\\var{radius1}$ and $\\var{radius2}$.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"radius1": {"name": "radius1", "group": "The circle", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "radius2": {"name": "radius2", "group": "The circle", "definition": "random(5..9)", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "The circle", "definition": "random(0..2)", "description": "Selector.
\n0 => 1 circle, >
\n1 => 1 circle, <
\n2 => 2 circles.
", "templateType": "anything", "can_override": false}, "eqn1": {"name": "eqn1", "group": "The circle", "definition": "simplify(expression(\"x^2 + y^2 > \" + radius1*radius1),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "eqn2": {"name": "eqn2", "group": "The circle", "definition": "simplify(expression(\"x^2 + y^2 < \" + radius2*radius2),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "centrex": {"name": "centrex", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "centrey": {"name": "centrey", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "incolor": {"name": "incolor", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "outcolor": {"name": "outcolor", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "radius": {"name": "radius", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "xmax": {"name": "xmax", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "xmin": {"name": "xmin", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "ymax": {"name": "ymax", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "ymin": {"name": "ymin", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["centrex", "centrey", "incolor", "outcolor", "radius", "xmax", "xmin", "ymax", "ymin"], "variable_groups": [{"name": "The circle", "variables": ["radius1", "radius2", "idx", "eqn1", "eqn2"]}, {"name": "The graph", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "2.6.7 Circle inequality no 2", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given a circle centre and radius, write an appropriate inequality for the region either inside or outside the circle.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "State an inequality that describes the points that lie inside outside the circle of radius $\\var{radius}$ with centre $(\\var{centrex}, \\var{centrey})$.
", "advice": "The points outside the circle are the points that are further from the centre than the distance of the radius.
\nThe points inside the circle are the points that are closer to the centre than the distance of the radius.
\nThus, putting the centre coordinates and the radius into the circle equation and then writing the inequality gives:
\n$\\var{inequality}$
\nThe diagram shows the area that satisfies the inequality shaded in cyan.
\n{correctgraph}
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\n0 => >
\n1 => <
", "templateType": "anything", "can_override": false}, "correctgraph": {"name": "correctgraph", "group": "The graph", "definition": "jessiecode(\n 400,400,[{xmin},{ymax},{xmax},{ymin}],\"incolor={incolor};outcolor={outcolor};xmin={xmin};ymin={ymin};xmax={xmax};ymax={ymax};R={radius};Cx={centrex};Cy={centrey};\"+safe(\n \"\"\"\n p=polygon([xmin,ymin],[xmin,ymax],[xmax,ymax],[xmax,ymin]) <Given a circle centre and a point on the circumference, write down the circle equation.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Obtain the equation of the circle which has centre $(\\var{centrex},\\var{centrey})$ and which passes through the point $(\\var{pointx},\\var{pointy})$.
", "advice": "We need to compute the distance from the centre of the circle to the point on the circumference. This distance is the radius, $r$.
\n$r=\\sqrt{\\left[(\\var{centrex})-(\\var{pointx})\\right]^2 + \\left[(\\var{centrey})-(\\var{pointy})\\right]^2}$
\n$r=\\sqrt{\\var{radiussquared}}$ $=\\var{radius}$
\nHence the circle equation is $\\var{equation}$.
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"centrex": {"name": "centrex", "group": "Ungrouped variables", "definition": "random(-9,9)", "description": "", "templateType": "anything", "can_override": false}, "centrey": {"name": "centrey", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "pointx": {"name": "pointx", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "pointy": {"name": "pointy", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "templateType": "anything", "can_override": false}, "radiussquared": {"name": "radiussquared", "group": "Ungrouped variables", "definition": "(centrex-pointx)^2+(centrey-pointy)^2", "description": "", "templateType": "anything", "can_override": false}, "radius": {"name": "radius", "group": "Ungrouped variables", "definition": "simplify(expression(\"sqrt(\"+radiussquared+\")\"),\"sqrtSquare\")", "description": "", "templateType": "anything", "can_override": false}, "intradius": {"name": "intradius", "group": "Ungrouped variables", "definition": "floor(eval(radius))", "description": "", "templateType": "anything", "can_override": false}, "evalradius": {"name": "evalradius", "group": "Ungrouped variables", "definition": "eval(radius)", "description": "", "templateType": "anything", "can_override": false}, "equation": {"name": "equation", "group": "Ungrouped variables", "definition": "simplify(expression(\"(x-\" + centrex + \")^2+(y-\"+ centrey + \")^2 = \" + radiussquared ),\"all\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "centrex<>pointx or centrey<>pointy", "maxRuns": 100}, "ungrouped_variables": ["centrex", "centrey", "pointx", "pointy", "radiussquared", "radius", "intradius", "evalradius", "equation"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{equation}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "2.6.9 Circle proof", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Fixed question: Given two points at opposite ends of a diameter, write down the equation of the circle.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Show that if $A(x_1, y_1)$ and $B(x_2, y_2)$ are at opposite ends of a diameter of a circle then the equation of the circle is $(x − x_1)(x − x_2) + (y − y_1)(y − y_2) = 0$.
\nHint: if $P$ is any point on the circle, obtain the slopes of the lines $AP$ and $BP$ and recall that the angle in a semicircle must be a right angle.
", "advice": "Let $P (x,y)$ be an arbitrary point on the circle.
\nThe the angle $\\angle APB$ is a right angle, meaning that $AP$ and $BP$ are perpendicular, meaning that the product of their gradients is $-1$.
\nLet $m_1$ be the gradient of $AP$ and let $m_2$ be the gradient of $BP$.
\nThen
\n$m_1 = \\dfrac{y-y_1}{x-x_1}\\quad$ and $\\quad m_2=\\dfrac{y-y_2}{x-x_2}$
\nSo
\n$m_1 m_2 = -1$
\n$ \\dfrac{y-y_1}{x-x_1} \\times \\dfrac{y-y_2}{x-x_2}=-1$
\nMultiplying through by the denominator:
\n$(y-y_1)(y-y_2) = -(x-x_1)(x-x_2)$
\ni.e.
\n$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$
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", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "State the equation of the unique circle which touches the $x$−axis at the point $(\\var{Cx},0)$ and which passes through the point $(\\var{px}, \\var{py})$.
", "advice": "Since the circle touches the $x$-axis at $(\\var{a},0)$, this means that this must be the lowest point of the circle. Hence the centre must be on the line $x=\\var{a}$, and the $y-$value at the centre must be equal to the radius.
\nHence the circle equation is
\n$(\\var{simplify(expression(\"x-\"+a),\"basic\")})^2+(y-r)^2=r^2$
\nand $(\\var{px},\\var{py})$ satisfies this equation:
\n$(\\var{simplify(expression(px+\"-\"+a),\"basic\")})^2+(\\var{py}-r)^2=r^2$
\nExpanding:
\n$\\var{(px-a)*(px-a)} + \\var{py*py} - \\var{2*py}r + r^2 = r^2$
\nRearranging:
\n$\\var{2*py}r = \\var{(px-a)*(px-a)+py*py}$
\n$r=\\var[fractionNumbers]{R}$
\nHence the equation of the circle is
\n$(\\var{simplify(expression(\"x-\"+a),\"basic\")})^2+(y-\\var{R})^2=\\var[fractionNumbers]{R*R}$
\n{correctgraph}
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"equation": {"name": "equation", "group": "The Circle", "definition": "expression(\"(x-\"+Cx+\")^2+(y-\"+Cy+\")^2=\"+R*R)", "description": "", "templateType": "anything", "can_override": false}, "Cx": {"name": "Cx", "group": "The Circle", "definition": "a", "description": "centre x-coordinate
", "templateType": "anything", "can_override": false}, "Cy": {"name": "Cy", "group": "The Circle", "definition": "R", "description": "centre y-value
", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "The Circle", "definition": "(1+py)/2", "description": "radius
", "templateType": "anything", "can_override": false}, "xminusa": {"name": "xminusa", "group": "The Circle", "definition": "random([-4,-3,-2,-1,1,2,3,4])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "The Circle", "definition": "random(-4..4)", "description": "", "templateType": "anything", "can_override": false}, "px": {"name": "px", "group": "The Circle", "definition": "xminusa + a", "description": "the x-coordinate of the point
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", "templateType": "anything", "can_override": false}, "playgraph": {"name": "playgraph", "group": "The graph", "definition": "jessiecode(\n 400,400,[{xmin},{ymax},{xmax},{ymin}],\"Cx={Cx};Cy={Cy};px={px};py={py};ax={a};\"+safe(\n \"\"\"\n Cx=0; Cy=0;\n P1x = 1;\n C=point(Cx,Cy) <You can move the black points on this graph to get an idea of where the circle needs to go.
\n{playgraph}
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\nQuestions generally have multiple versions, clicking the \"Try another question like this one\" button will generate a new version.
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