// Numbas version: finer_feedback_settings
{"name": "2.6.9 Circle proof", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2.6.9 Circle proof", "tags": [], "metadata": {"description": "<p>Fixed question: Given two points at opposite ends of a diameter, write down the equation of the circle.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>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 $(x &minus; x_1)(x &minus; x_2) + (y &minus; y_1)(y &minus; y_2) = 0$.</p>\n<p>Hint: 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.</p>", "advice": "<p>Let $P (x,y)$ be an arbitrary point on the circle.</p>\n<p>The the angle $\\angle APB$ is a right angle, meaning that $AP$ and $BP$ are perpendicular, meaning that the product of their gradients is $-1$.</p>\n<p>Let $m_1$ be the gradient of $AP$ and let $m_2$ be the gradient of $BP$.</p>\n<p>Then</p>\n<p style=\"padding-left: 40px;\">$m_1 = \\dfrac{y-y_1}{x-x_1}\\quad$ and $\\quad m_2=\\dfrac{y-y_2}{x-x_2}$</p>\n<p>So</p>\n<p style=\"padding-left: 80px;\">$m_1 m_2 = -1$</p>\n<p style=\"padding-left: 40px;\">$ \\dfrac{y-y_1}{x-x_1} \\times \\dfrac{y-y_2}{x-x_2}=-1$</p>\n<p>Multiplying through by the denominator:</p>\n<p style=\"padding-left: 40px;\">$(y-y_1)(y-y_2) = -(x-x_1)(x-x_2)$</p>\n<p>i.e.</p>\n<p style=\"padding-left: 40px;\">$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "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", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}