[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-574154-105":53,"doc-detail-574154-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","conic-sections-formulas-reference-guide","Conic Sections Formulas - Reference Guide","","This comprehensive reference document outlines the essential mathematical formulas for analyzing conic sections, including parabolas, ellipses, and hyperbolas. It provides a structured breakdown of equations, vertices, foci, axes, and directrix properties for both vertical and horizontal orientations. Additionally, the document features a classification table based on the general quadratic equation format Ax2+Cy2+Dx+Ey+F=0, allowing users to quickly identify the type of conic section based on coefficient relationships. This guide serves as a practical tool for students and educators in geometry and calculus to verify properties and understand the algebraic representation of these fundamental curves. Its clear tabular layout facilitates quick navigation and application of complex coordinate geometry concepts in various academic or technical settings.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/conic-sections-formulas-reference-guide/574154/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/conic-sections-formulas-reference-guide/574154.png","ImageObject",442,249,{"name":88,"@type":89},"Jacob","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-14",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you distinguish between conic sections using the general equation Ax2+Cy2+Dx+Ey+F=0?","Question",{"text":108,"@type":109},"You can classify them by examining the coefficients A and C: circles have A=C, parabolas have AC=0 (where both are not zero), ellipses have AC>0, and hyperbolas have AC\u003C0.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the relationship between parameters a, b, and c in an ellipse?",{"text":113,"@type":109},"For an ellipse, the parameters are related by the equation a2=b2+c2, where 'a' represents the length related to the major axis and 'b' represents the length related to the minor axis.",{"name":115,"@type":106,"acceptedAnswer":116},"What is the equation for the directrix of a horizontal parabola?",{"text":117,"@type":109},"The equation for the directrix of a horizontal parabola is x=h-p, where (h,k) is the vertex and p is the parameter determining the direction and width of the opening.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},574154,1792086771,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":4},962084931830,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Conic Sections Formulas  \n\n| Parabola |  |  |\n| --- | --- | --- |\n|  | Vertical Axis | Horizontal axis |\n| equation | (x-h)2=4p(y-k) | (y-k)2=4p(x-h) |\n| Axis of symmetry | x=h | y=k |\n| Vertex | (h,k) | (h,k) |\n| Focus | (h,k+p) | (h+p,k) |\n| Directrix | y=k-p | x=h-p |\n| Direction of opening | p>0 then up; p\u003C0 then down | p>0 then rignt; p\u003C0 then left |\n\n| Ellipse |  |  |\n| --- | --- | --- |\n|  | Vertical Major Axis | Horizontal Major axis |\n| equation | 􀀋x~~ ~~2h􀀌2~~ ~~ 􀀎 􀀋y~~ ~~2k~~ ~~􀀌2~~ ~~ 􀀠 1 | 􀀋x~~ ~~2h􀀌2~~ ~~ 􀀎 􀀋y~~ ~~2k~~ ~~􀀌2~~ ~~ 􀀠 1 |\n| center | (h,k) | (h,k) |\n| Vertices | (h,k±a) | (h±a,k) |\n| Foci | (h,k±c) | (h±c,k) |\n| Major axis equation | 2a=length of major axis |  |\n| Minor axis equation | 2b=length of minor axis |  |\n| Equation that relates a, b, and c | a2=b2+c2 |  |\n| Eccentricity of an ellipse | e=(c/a) |  |\n\n| Hyperbola |  |  |\n| --- | --- | --- |\n|  | Vertical Transverse Axis | Horizontal Transverse axis |\n| equation | 􀀋y~~ ~~2k~~ ~~􀀌2~~ ~~ 􀀐 􀀋x~~ ~~2h􀀌2~~ ~~ 􀀠 1 | 􀀋x~~ ~~2h􀀌2~~ ~~ 􀀐 􀀋y~~ ~~2k~~ ~~􀀌2~~ ~~ 􀀠 1 |\n| center | (h,k) | (h,k) |\n| Vertices | (h,k±a) | (h±a,k) |\n| Foci | (h,k±c) | (h±c,k) |\n| Assymptote equation | y 􀀠 k 􀁲 ab 􀀋x 􀀐 h􀀌 | y 􀀠 k 􀁲 ba 􀀋x 􀀐 h􀀌 |\n| Equation relating a, b, and c | c2=a2+b2 |  |\n\n\n| Classifying conic sections | Circles | Parabola | Ellipse | Hyperbola |\n| --- | --- | --- | --- | --- |\n| Ax2+Cy2+Dx+Ey+F=0 | A=C | AC=0, Both\u003Cbr>are not 0 | AC>0 | AC\u003C0 |\n\nSources: [www.teacherweb.com/IL/HiawathaSchools/.../](www.teacherweb.com/IL/HiawathaSchools/.../Conicsectionsformulasheet.doc)[Conicsectionsformulas](www.teacherweb.com/IL/HiawathaSchools/.../Conicsectionsformulasheet.doc)[heet.doc](www.teacherweb.com/IL/HiawathaSchools/.../Conicsectionsformulasheet.doc)[ ](www.teacherweb.com/IL/HiawathaSchools/.../Conicsectionsformulasheet.doc)[www.StewartCalculus.com](www.StewartCalculus.com)","cbCaivCrkC88EnCV","https://ap.wps.com/l/cbCaivCrkC88EnCV","pdf",73220,"English","# Parabola\n## Vertical and Horizontal Axes\n# Ellipse\n## Major and Minor Axes\n# Hyperbola\n## Transverse Axes and Asymptotes\n# Classifying Conic Sections\n## General Form Coefficients","[{\"question\":\"How do you distinguish between conic sections using the general equation Ax2+Cy2+Dx+Ey+F=0?\",\"answer\":\"You can classify them by examining the coefficients A and C: circles have A=C, parabolas have AC=0 (where both are not zero), ellipses have AC\\u003e0, and hyperbolas have AC\\u003c0.\"},{\"question\":\"What is the relationship between parameters a, b, and c in an ellipse?\",\"answer\":\"For an ellipse, the parameters are related by the equation a2=b2+c2, where 'a' represents the length related to the major axis and 'b' represents the length related to the minor axis.\"},{\"question\":\"What is the equation for the directrix of a horizontal parabola?\",\"answer\":\"The equation for the directrix of a horizontal parabola is x=h-p, where (h,k) is the vertex and p is the parameter determining the direction and width of the opening.\"}]","Conic Sections Formulas - Reference Guide | PDF",1791985242]