[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-574152-105":53,"doc-detail-574152-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-formula-sheet","Conic Sections - Formula Sheet","","This comprehensive reference document serves as a mathematical guide for conic sections, encompassing essential formulas for analytical geometry. It details fundamental equations for the distance, midpoint, slope, and point-slope forms. Furthermore, it provides structural breakdowns and standard forms for parabolas, ellipses, hyperbolas, and circles, specifying their vertices, foci, centers, and directrices. The document also outlines the discriminant method for identifying conic sections based on the general quadratic equation coefficients, making it a valuable tool for students and practitioners in algebra and geometry to quickly verify properties and standard forms of conic curves.",{"@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-formula-sheet/574152/",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-formula-sheet/574152.png","ImageObject",442,249,{"name":88,"@type":89},"Jake","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-16","2026-10-14",true,{"@type":98,"interactionType":99,"userInteractionCount":73},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you identify a circle based on the general quadratic equation Ax²+Bxy+Cy²+Dx+Ey+F=0?","Question",{"text":108,"@type":109},"A conic section is a circle if the discriminant B²-4AC is less than zero, the coefficient B equals zero, and A equals C.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What is the standard equation for a parabola opening vertically?",{"text":113,"@type":109},"The equation for a vertical parabola is (x-h)² = 4p(y-k), where (h,k) is the vertex and p determines the direction of opening.",{"name":115,"@type":106,"acceptedAnswer":116},"How is the focal distance 'c' calculated for an ellipse and a hyperbola?",{"text":117,"@type":109},"For an ellipse, c² = a² - b². For a hyperbola, the relationship is c² = a² + b².","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},574152,1792110396,{"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":73,"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},962084928904,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Conic Section Formula Sheet  \nGeneral Formulas  \nMidpoint  \n⎛ x 1 + x2 y 1 + y2  ⎞  \n⎝ ⎜ 2 , 2 ⎟⎠  \nDistance  \n(x2 − x1 )2 + (y2 − y1 )2  \nSlope  Slope-Intercept  \ny2 − y 1  y = mx + b  \nx2 − x 1  \nPoint-slope  \ny − y1 = m (x − x1  \n)  \n|  | Parabola |\n| --- | --- |\n| Vertical: Opens up or down | Horizontal: Opens right or left |\n| Equation: (x − h)2 = 4p ( y − k) | Equation: ( y − k)2 = 4p (x − h) |\n| Vertex: (h,k) | Vertex: (h,k) |\n| Focus: (h, k + p) | Focus: (h + p, k) |\n| Directrix: y = k-p | Directrix: x = h-p |\n| Axis of Symetry: x = h | Axis of Symetry: y = k |\n| Ifp > 0 parabola opens up, ifp \u003C 0 parabola opens down | Ifp > 0 parabola opens right, ifp \u003C 0 parabola opens left |\n| Horizontal: Major Axis Parallel to x-axis | Ellipse\u003Cbr>Vertical: Major Axis Parallel to y-axis |\n| (x − h)2 ( y − k)2 Equation:  a2  +  b2  = 1 | (x − h)2 ( y − k)2\u003Cbr>Equation:  b2  +  a2  = 1 |\n| Center: (h,k) | Center: (h,k) |\n| Vertices: (h ± a, k) | Vertices: (h, k ± a) |\n| Co-vertices: (h, k ± b) | Co-vertices: (h ± b, k) |\n| Foci: (h ± c, k); where c2 = a2-b2 | Foci: (h, k ± c); where c2 = a2-b2 |\n| Horizontal: Transverse Axis Parallel to x-axis | Hyperbola\u003Cbr>Vertical: Transverse Axis Parallel to y-axis |\n| (x − h)2 ( y − k)2 Equation:  a2  −  b2  = 1 | ( y − k)2 (x − h)2\u003Cbr>Equation:  a2  −  b2  = 1 |\n| Center: (h,k) | Center: (h,k) |\n| Vertices: (h ± a, k)\u003Cbr>Foci: (h ± c, k); where c2 = a2 + b2 | Vertices: (h, k ± a)\u003Cbr>Foci: (h, k ± c); where c2 = a2 + b2 |\n| Asymptotes: y = ± b (x − h)+ k\u003Cbr>a | Asymptotes: y = ± ab(x − h)+ k |\n| Equation: (x − h)2 + ( y − k)2 = r 2 | Circle\u003Cbr>Center: (h,k)\u003Cbr>Radius: r |\n\nIdentifying Conic Sections  \nTo identify conic sections of the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, and C do not all equal zero:  \nCircle: If B2-4AC \u003C 0, with B = 0 andA = C  \nEllipse: If B2-4AC \u003C 0, with B ≠ 0 or A ≠ C  \nHyperbola: If B2-4AC > 0  \nParabola: If B2-4AC = 0","cbCaio8bJW9oY86t","https://ap.wps.com/l/cbCaio8bJW9oY86t","pdf",110006,"English","# General Formulas\n# Parabola\n# Ellipse\n# Hyperbola\n# Circle\n# Identifying Conic Sections","[{\"question\":\"How do you identify a circle based on the general quadratic equation Ax²+Bxy+Cy²+Dx+Ey+F=0?\",\"answer\":\"A conic section is a circle if the discriminant B²-4AC is less than zero, the coefficient B equals zero, and A equals C.\"},{\"question\":\"What is the standard equation for a parabola opening vertically?\",\"answer\":\"The equation for a vertical parabola is (x-h)² = 4p(y-k), where (h,k) is the vertex and p determines the direction of opening.\"},{\"question\":\"How is the focal distance 'c' calculated for an ellipse and a hyperbola?\",\"answer\":\"For an ellipse, c² = a² - b². For a hyperbola, the relationship is c² = a² + b².\"}]","Conic Sections - Formula Sheet | PDF",1791985237]