foci 16x^2+25y^2=400

{ "query": { "display": "foci $$16x^{2}+25y^{2}=400$$", "symbolab_question": "CONIC#foci 16x^{2}+25y^{2}=400" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Ellipse", "subTopic": "foci", "default": "(3,0),(-3,0)" }, "steps": { "type": "interim", "title": "Ellipse foci given $$16x^{2}+25y^{2}=400:{\\quad}\\left(3,\\:0\\right),\\:\\left(-3,\\:0\\right)$$", "steps": [ { "type": "definition", "title": "Ellipse Foci", "text": "For an ellipse with major axis parallel to the x-axis, the Foci (focus points) are defined as $$\\left(h+c,\\:k\\right),\\:\\left(h-c,\\:k\\right),\\:$$<br/> where $$c=\\sqrt{a^2-b^2}\\:$$ is the distance from the center $$\\left(h,\\:k\\right)\\:$$to a focus" }, { "type": "step", "result": "\\left(h+c,\\:k\\right),\\:\\left(h-c,\\:k\\right)" }, { "type": "step", "primary": "Calculate ellipse properties" }, { "type": "interim", "title": "$$16x^{2}+25y^{2}=400:{\\quad}$$Ellipse with center $$\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4$$", "input": "16x^{2}+25y^{2}=400", "steps": [ { "type": "definition", "title": "Ellipse standard equation", "text": "$$\\frac{\\left(x-h\\right)^{2}}{a^2}+\\frac{\\left(y-k\\right)^{2}}{b^2}=1\\:$$is the ellipse standard equation<br/>with center $$\\left(h,\\:k\\right)\\:$$and $$a,\\:b$$ are the semi-major and semi-minor axes" }, { "type": "interim", "title": "Rewrite $$16x^{2}+25y^{2}=400\\:$$in the form of the standard ellipse equation", "input": "16x^{2}+25y^{2}=400", "steps": [ { "type": "step", "primary": "Divide by coefficient of square terms: $$16$$", "result": "x^{2}+\\frac{25}{16}y^{2}=25" }, { "type": "step", "primary": "Divide by coefficient of square terms: $$25$$", "result": "\\frac{1}{25}x^{2}+\\frac{1}{16}y^{2}=1" }, { "type": "step", "primary": "Refine", "result": "\\frac{x^{2}}{25}+\\frac{y^{2}}{16}=1" }, { "type": "step", "primary": "Rewrite in standard form", "result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{4^{2}}=1" } ], "meta": { "interimType": "Ellipse Canonical Format 1Eq" } }, { "type": "step", "result": "\\frac{\\left(x-0\\right)^{2}}{5^{2}}+\\frac{\\left(y-0\\right)^{2}}{4^{2}}=1" }, { "type": "step", "primary": "Therefore ellipse properties are:", "result": "\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4" }, { "type": "step", "primary": "$$a>b\\:$$therefore $$a\\:$$is semi-major axis and $$b\\:$$is semi-minor axis", "result": "\\mathrm{Ellipse\\:with\\:center}\\:\\left(h,\\:k\\right)=\\left(0,\\:0\\right),\\:a=5,\\:b=4" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "\\left(0+c,\\:0\\right),\\:\\left(0-c,\\:0\\right)" }, { "type": "step", "primary": "Compute $$c:$$" }, { "type": "interim", "title": "$$c=\\sqrt{5^{2}-4^{2}}:{\\quad}3$$", "input": "\\sqrt{5^{2}-4^{2}}", "steps": [ { "type": "step", "primary": "$$5^{2}=25$$", "result": "=\\sqrt{25-4^{2}}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{25-16}" }, { "type": "step", "primary": "Subtract the numbers: $$25-16=9$$", "result": "=\\sqrt{9}" }, { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74f6cUYKncTCZw7Lf+vpnnl1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPIFNZvSS5A2llKcsOioUT3tE+QZDw8sNbXVMhDKVB/Cmpw8B6izv73gtWusAUWvvnM" } }, { "type": "step", "result": "\\left(0+3,\\:0\\right),\\:\\left(0-3,\\:0\\right)" }, { "type": "step", "primary": "Refine", "result": "\\left(3,\\:0\\right),\\:\\left(-3,\\:0\\right)" } ], "meta": { "solvingClass": "Ellipse" } }, "plot_output": { "meta": { 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