[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-574155-105":3,"detail-sidebar-cat-1-en-105":81,"doc-detail-574155-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","conic-sections-cheat-sheet-quick-reference","Conic Sections Cheat Sheet - Quick Reference","","A concise cheat sheet summarizing conic sections with standard coordinate forms and key geometric parameters. It covers parabolas using vertex, focus, and directrix relationships, plus key properties of points equidistant from focus and directrix. It also provides organized formulas for circles/ellipses and hyperbolas, including center-based equations, foci/vertices, and hyperbola asymptotes. Intended for fast review of core analytic geometry results.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/conic-sections-cheat-sheet-quick-reference/574155/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/conic-sections-cheat-sheet-quick-reference/574155.png","ImageObject",442,249,{"name":42,"@type":43},"Connor ","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-16","2026-10-14",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the parabola equations in terms of vertex and focus/directrix parameters?","Question",{"text":63,"@type":64},"The sheet states parabolas with vertex (h, k) and gives forms using shifts like (x-h) and (y-k), plus focus and directrix relationships. It also notes the point property equidistant from the focus and directrix.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are circles and ellipses represented when their center is (h, k)?",{"text":68,"@type":64},"It provides center-based formulas for circles and ellipses, including the circle equation (with r) and ellipse equations using a and b. It also lists focal coordinates and vertex coordinates for wide/tall ellipse cases.",{"name":70,"@type":61,"acceptedAnswer":71},"What key features of hyperbolas does the cheat sheet include?",{"text":72,"@type":64},"It includes hyperbola standard forms for horizontal and vertical orientations, relationships for c² in terms of a² and b², and the coordinates of foci and vertices. It also states asymptote equations and the constant difference of focal distances property.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},574155,1792011645,{"code":4,"msg":82,"data":83},"success",[84,89,94,99,104,109,114,119,123],{"id":85,"doc_module":22,"doc_module_name":25,"category_name":86,"show_sort_weight":87,"slug":88},11,"Presentations",90,"presentations",{"id":90,"doc_module":22,"doc_module_name":25,"category_name":91,"show_sort_weight":92,"slug":93},12,"Resumes",80,"resumes",{"id":95,"doc_module":22,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},14,"Invoices",70,"invoices",{"id":100,"doc_module":22,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},15,"Posters",60,"posters",{"id":105,"doc_module":22,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},16,"Social Media",50,"social-media",{"id":110,"doc_module":22,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},17,"Forms",40,"forms",{"id":115,"doc_module":22,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},18,"Letters",30,"letters",{"id":120,"doc_module":22,"doc_module_name":25,"category_name":121,"show_sort_weight":55,"slug":122},21,"Paper Templates","papers-templates",{"id":124,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":125},158,"general-158",{"code":4,"msg":82,"data":127},{"doc_id":79,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":124,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":22,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":4},687207022233,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Conic Sections Cheat Sheet  \nby CROSSANT (CROSSANT) [via](via cheatography.com/186482/cs/38990/)[ cheatography.com/186482/cs/38990/](via cheatography.com/186482/cs/38990/)  \nParabolas with vertex (h, k)  \n2  \nOpening up/down (x-h) = ±4p(y-k)  \nVertical Focus (h , k+p)  \nDirectrix y=k-p  \n2  \nOpening right/left (y-k) = ±4p(x-h)  \nHorizontal Focus (h+p , k)  \nDirectrix x = h-p  \nAny point on a parabola is equidistant from the parabola's focus and directrix  \nConic Cross-Sections Diagram  \nParabola opening upwards  \nCircles/Ellipses with center (h, k)  \n| Circle | 2 2 2 (x-h) + (y-k) = r |\n| --- | --- |\n| Circle Focus | (h , k) |\n| Circle Vertices | None |\n| Wide Ellipse | 2 2 2 2 (x-h) /a + (y-k) /b = 1 |\n| Wide Foci | (h±c , k) |\n| Wide Vertices | (h±a , k±b) |\n| Tall Ellipse | 2 2 2 2 (x-h) /b + (y-k) /a = 1 |\n| Tall Foci | (h , k±c) |\n\nBy CROSSANT (CROSSANT)  \n[cheatography.com/crossant/](cheatography.com/crossant/)  \nCircles/Ellipses with center (h, k) (cont)  \n| Tall Vertices | (h±b , k±a) |\n| --- | --- |\n\nc² = a² -b² and a > b > 0  \nFormulas for foci generate two different points (+c and-c) , and formulas for vertices generate four different vertices: (h+a , k) (ha , k) (h , k+b) and (h , k-b)  \nDistances between a focal point to any point on the ellipse , plus the distance of the other focal point to that same point on the ellipse , gives a sum of distances that is constant for any point on the ellipse  \nWide Ellipse  \nHyperbolas with center (h, k)  \nBranches opening horizontally  \n2 2  \n(x-h) /a - (y- 2 2  \nk) /b = 1  \nHorizontal Foci (h±c , k)  \nHorizontal Vertices (h±a , k)  \nAsymptotes Branches opening vertically  \ny-k = ±(b/a)(x-h)  \n2 2  \n(y-k) /a - (x- 2 2  \nh) /b = 1  \nVertical Foci (h , k±c)  \nVertical Vertices (h , k±a)  \nPublished 28th May , 2023. Last updated 22nd April , 2026. Page 1 of 1.  \nHyperbolas with center (h, k) (cont)  \n| Asymptotes | y-k=±(a/b)(x-h) |\n| --- | --- |\n\nc² = a² + b² and a , b ≠ 0  \nFormulas for foci generate two different points (+c and-c) , formulas for vertices generate two different points (+a and-a) , and formulas for asymptotes generate two different asymptotes (+(a/b) and-(a/b) or +(b/a) and-(b/a))  \nDistance of a focal point to a point on either hyperbola branch , minus distance of the other focal point to that same point on that same hyperbola branch , gives a value whose magnitude is constant for any point on either hyperbola branch  \nHorizontal Hyperbola  \nHorizontal Hyperbola Asymptotes  \nSponsored by ApolloPad.com Everyone has a novel in them. Finish Yours!  \n[https://apollopad.com](https://apollopad.com)","cbCait6c6rTcbtbr","https://ap.wps.com/l/cbCait6c6rTcbtbr","pdf",199028,"English","# Parabolas\n# Conic Cross-Sections Diagram\n# Circles and Ellipses\n# Hyperbolas","[{\"question\":\"What are the parabola equations in terms of vertex and focus/directrix parameters?\",\"answer\":\"The sheet states parabolas with vertex (h, k) and gives forms using shifts like (x-h) and (y-k), plus focus and directrix relationships. It also notes the point property equidistant from the focus and directrix.\"},{\"question\":\"How are circles and ellipses represented when their center is (h, k)?\",\"answer\":\"It provides center-based formulas for circles and ellipses, including the circle equation (with r) and ellipse equations using a and b. It also lists focal coordinates and vertex coordinates for wide/tall ellipse cases.\"},{\"question\":\"What key features of hyperbolas does the cheat sheet include?\",\"answer\":\"It includes hyperbola standard forms for horizontal and vertical orientations, relationships for c² in terms of a² and b², and the coordinates of foci and vertices. It also states asymptote equations and the constant difference of focal distances property.\"}]","Conic Sections Cheat Sheet - Quick Reference | PDF",1791985243]