Recognising the equation of a circle.
"}, "tags": [], "functions": {}, "name": "Circles - recognizing the equation", "ungrouped_variables": ["rs", "ccx", "ccy", "xmult", "ymult", "ax", "bx", "ay", "by"], "parts": [{"variableReplacementStrategy": "originalfirst", "distractors": ["", "", "This is a quadratic equation", "This is the equation of an ellipse", "", "This is the equation of an ellipse", "This is the equation of a hyperbola"], "unitTests": [], "showFeedbackIcon": false, "variableReplacements": [], "marks": 0, "extendBaseMarkingAlgorithm": true, "maxMarks": 0, "maxAnswers": 0, "type": "m_n_2", "matrix": ["4/4", "4/4", "-3/4", "-3/4", "4/4", "-3/4", "-3/4"], "displayColumns": "1", "warningType": "none", "showCorrectAnswer": true, "scripts": {}, "displayType": "checkbox", "customMarkingAlgorithm": "", "showCellAnswerState": true, "choices": ["$\\simplify[all]{(x-{ccx})^2+(y-{ccy})^2={rs[0]}}$
", "$\\simplify[all]{x^2+(y-{ccy})^2={rs[1]}}$
", "$\\simplify[all]{(x-{ccx})^2={rs[2]}}$
", "$\\simplify[all]{{xmult}(x-{ccx})^2+{ymult}(y-{ccy})^2={rs[3]}}$
", "$\\simplify[all]{{ax}x^2+{bx}x+{ax}y^2+{by}y={rs[4]}}$
", "$\\simplify[all]{{ax}x^2+{bx}x+{ay}y^2+{by}y={rs[5]}}$
", "$\\simplify[all]{x^2+{bx}x-y^2+{by}y={rs[5]}}$
"], "minAnswers": 0, "shuffleChoices": true, "minMarks": 0}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"ccy": {"description": "", "templateType": "anything", "definition": "random(-12..12)", "name": "ccy", "group": "Ungrouped variables"}, "by": {"description": "", "templateType": "anything", "definition": "random(-6..6)", "name": "by", "group": "Ungrouped variables"}, "ymult": {"description": "", "templateType": "anything", "definition": "random(1..12 except xmult)", "name": "ymult", "group": "Ungrouped variables"}, "ccx": {"description": "", "templateType": "anything", "definition": "random(-12..12 except 0)", "name": "ccx", "group": "Ungrouped variables"}, "ay": {"description": "", "templateType": "anything", "definition": "random(1..6 except ax)", "name": "ay", "group": "Ungrouped variables"}, "xmult": {"description": "", "templateType": "anything", "definition": "random(1..12)", "name": "xmult", "group": "Ungrouped variables"}, "rs": {"description": "", "templateType": "anything", "definition": "repeat(random(1..60),6)", "name": "rs", "group": "Ungrouped variables"}, "ax": {"description": "", "templateType": "anything", "definition": "random(1..6)", "name": "ax", "group": "Ungrouped variables"}, "bx": {"description": "", "templateType": "anything", "definition": "random(-6..6)", "name": "bx", "group": "Ungrouped variables"}}, "extensions": [], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "The standard form equation of a circle is $(x-a)^2+(y-b)^2=r^2$. It represents a circle of radius $r$ with centre $(a,b)$.
\nIf an equation is written in the form $cx^2+dx+ey^2+fy+g=0$, where $c,d,e,f,g$ are numbers, it could be a circle, but it might be something else (for example, an ellipse, a hyperbola, or even an empty graph). If the coefficients of $x^2$ and $y^2$ are the same then the equation is of a circle. These values being the same allow us to divide by that coefficient and start completing the square to get the equation in to the standard form of a circle.
", "statement": "Which of the following are equations of a circle? (This question employs negative marking, therefore marks are deducted if you select an incorrect answer.)
", "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}