y^2+8x+10y-5=0

{ "query": { "display": "$$y^{2}+8x+10y-5=0$$", "symbolab_question": "CONIC#y^{2}+8x+10y-5=0" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(\\frac{15}{4},-5),p=-2" }, "steps": { "type": "interim", "title": "$$y^{2}+8x+10y-5=0:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(\\frac{15}{4},\\:-5\\right),\\:$$and focal length $$|p|=2$$", "input": "y^{2}+8x+10y-5=0", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(x-h\\right)=\\left(y-k\\right)^{2}\\:$$ is the standard equation for a right-left facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$y^{2}+8x+10y-5=0\\:$$in the standard form", "input": "y^{2}+8x+10y-5=0", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "8x=-y^{2}-10y+5" }, { "type": "step", "primary": "Divide by $$8$$", "result": "x=-\\frac{y^{2}}{8}-\\frac{5y}{4}+\\frac{5}{8}" }, { "type": "interim", "title": "Complete the square $$-\\frac{y^{2}}{8}-\\frac{5y}{4}+\\frac{5}{8}:{\\quad}-\\frac{1}{8}\\left(y+5\\right)^{2}+\\frac{15}{4}$$", "input": "-\\frac{y^{2}}{8}-\\frac{5y}{4}+\\frac{5}{8}", "steps": [ { "type": "step", "primary": "Write $$-\\frac{y^{2}}{8}-\\frac{5y}{4}+\\frac{5}{8}\\:$$in the form: $$x^2+2ax+a^2$$", "secondary": [ "Factor out $$-\\frac{1}{8}$$" ], "result": "-\\frac{1}{8}\\left(y^{2}+10y-5\\right)" }, { "type": "interim", "title": "$$2a=10{\\quad:\\quad}a=5$$", "input": "2a=10", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2a=10", "result": "a=5", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2a}{2}=\\frac{10}{2}" }, { "type": "step", "primary": "Simplify", "result": "a=5" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7voy2Rve8s/Lw2EKnTt9DDpN1pXT08zEQpn0WJ6CFMXBgNJ67EOD8lvNAD3ifVxEo6ChE8r/ZN/+ka9wpb6F5VqD/ZippUbjkNaiuO9fSpCiJzb4WJ2gItQG093k/9yqQn0DOUCKhNs+Rsl0Db7zAuun+A9tiGSEHxUfFCM12u5XtBq87Zl3M67ytyMGVUEYcjAzZ8ohqFjLfdHD6GsAQJepcdHX1Q1LhJE1wNQB/28PkU5TkMMWES51yFsTkJT6lflB44+cAhbl05D7N1o4XnYUDxBuztWdWdEmOXBe3LLpWPNqzEhkP5qD6ioNU4s9fi9c/nHGhIqvPRpttdSi2lpJj/5xKKdB5FHwO4G/fHn8mxF5pxAejgMBdXf8NHl+mrwUyEiZKbMR7oqWL9N3ZCJoyhYICkHY06PRn2J5cYsZn/BYtBjyDqsh6dvoR+cjA" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Add and subtract $$5^{2}\\:$$", "result": "-\\frac{1}{8}\\left(y^{2}+10y-5+5^{2}-5^{2}\\right)" }, { "type": "step", "primary": "$$x^2+2ax+a^2=\\left(x+a\\right)^2$$", "secondary": [ "$$y^{2}+10y+5^{2}=\\left(y+5\\right)^{2}$$", "Complete the square" ], "result": "-\\frac{1}{8}\\left(\\left(y+5\\right)^{2}-5-5^{2}\\right)" }, { "type": "step", "primary": "Simplify", "result": "-\\frac{1}{8}\\left(y+5\\right)^{2}+\\frac{15}{4}" } ], "meta": { "solvingClass": "Equations", "interimType": "Complete Square 1Eq" } }, { "type": "step", "result": "x=-\\frac{1}{8}\\left(y+5\\right)^{2}+\\frac{15}{4}" }, { "type": "step", "primary": "Subtract $$\\frac{15}{4}$$ from both sides", "result": "x-\\frac{15}{4}=-\\frac{1}{8}\\left(y+5\\right)^{2}" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$-\\frac{1}{8}$$", "result": "-8\\left(x-\\frac{15}{4}\\right)=\\left(y+5\\right)^{2}" }, { "type": "step", "primary": "Rewrite in standard form", "result": "4\\left(-2\\right)\\left(x-\\frac{15}{4}\\right)=\\left(y-\\left(-5\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format 1Eq" } }, { "type": "step", "result": "4\\left(-2\\right)\\left(x-\\frac{15}{4}\\right)=\\left(y-\\left(-5\\right)\\right)^{2}" }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(\\frac{15}{4},\\:-5\\right),\\:p=-2" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4(-2)(x-\\frac{15}{4})}-5", "displayFormula": "4(-2)(x-\\frac{15}{4})=(y-(-5))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4(-2)(x-\\frac{15}{4})}-5", "displayFormula": "4(-2)(x-\\frac{15}{4})=(y-(-5))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{23}{4}", "displayFormula": "x=\\frac{23}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(\\frac{15}{4},-5)", "(\\frac{7}{4},-5)" ], "pointsDecimal": [ { "fst": 3.75, "snd": -5 }, { "fst": 1.75, "snd": -5 } ], "attributes": [ { "color": "PURPLE", "labels": [ 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