Given that the circle touches the x-axis at a given point and given a point on the circumference, find the equation of the circle.
", "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": {}, "extensions": ["jsxgraph"], "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
", "templateType": "anything", "can_override": false}, "py": {"name": "py", "group": "The Circle", "definition": "xminusa * xminusa", "description": "the y-coordinate of the point
", "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}
", "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", "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/"}]}