[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-574153-105":53,"doc-detail-574153-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-1-conic-sections-1-cheat-sheet","Conic Sections 1 - Conic Sections 1 Cheat Sheet","","Conic Sections 1 builds on year 1 graph sketching skills and introduces conics as intersections of planes with cones. The chapter explains parametric equations and how to convert them to Cartesian form by eliminating parameters, illustrated with a trigonometric identity example yielding a circle. It then covers parabolas, focusing on focus-directrix properties, parametric representations, and deriving focus and directrix from a parabola equation, followed by rectangular hyperbolas and their Cartesian/parametric forms. Finally, it develops tangent gradient methods using parametric or implicit differentiation.",{"@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-1-conic-sections-1-cheat-sheet/574153/",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-1-conic-sections-1-cheat-sheet/574153.png","ImageObject",442,249,{"name":88,"@type":89},"\tCallum ","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","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},"What is the relationship between conic sections and slicing cones?","Question",{"text":108,"@type":109},"Conic sections are graphs formed by the intersection of a plane with a cone. Different slices produce shapes such as circles, ellipses, parabolas, and hyperbolas.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How do you convert parametric equations to a Cartesian equation?",{"text":113,"@type":109},"Eliminate the parameter between the x- and y-parameter equations, often using substitution. Watch for trigonometric identities that can simplify the elimination.",{"name":115,"@type":106,"acceptedAnswer":116},"How are focus and directrix used to describe a parabola?",{"text":117,"@type":109},"A parabola is the locus of points whose distance from the focus equals the distance from the directrix. For a parabola in the form y^2 = 4ax, the focus is (a, 0) and the directrix is x = -a.","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},574153,1792103554,{"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},137451211410,"https://ap-avatar.wpscdn.com/avatar/2000bb0a9246f588df?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786362646172706240","Conic Sections 1 Cheat Sheet  \nThis chapter aims to build upon the graph sketching skills you developed in year 1.  \nAs described by the name, conic sections are planes produced by slicing two cones placed point to point-more mathematically, the intersection of cones with planes are the graphs that you will see in this chapter. Conic sections are used in modelling: planets travel in ellipses, the mirrors in solar power stations are parabolas and a variety of conic sections are used in engineering. In the set of diagrams to the right, diagram 1 shows how a circle is obtained from a conic section, 2 shows an ellipse, 3 shows a parabola and 4 shows a hyperbola.  \nParametric equations.  \nUnlike the Cartesian form of equations that you will have seen throughout school, parametric equations are given in terms of an independent variable, called a parameter-most of the time the parameter will be denoted ‘t’, and the equations are given in the form 􀝔 = 􀝌 (􀝐), 􀝕 = 􀝍 (􀝐) . By substituting in specific values of t you can find the co-ordinates of a point on the curve. To convert  \nparametric equations to Cartesian, you must eliminate the parameter between the equations-this can be often done by normal substitution, but keep an eye out for different trigonometric identities that can eliminate the parameter:  \nExample 1: A Curve has parametric equations 􀝔 = 􀝎sin(􀝐) and 􀝕 = 􀝎cos(􀝐), 􀝐 ∈ ℝ and r is a positive constant. Find the Cartesian equation of the curve.  \n| Notice that by squaring each equation, we can use the trigonometric identity sin2 􀟠 + cos2 􀟠 = 1. | 􀝔 2 = 􀝎 2sin2 􀝐, 􀝕 2 = 􀝎 2 cos2 􀝐 |\n| --- | --- |\n| Add the equations together to put in the form of the identity we’re using. | 􀝔 2 + 􀝕 2 = 􀝎 2 (sin2 􀝐 + cos2 􀝐)\u003Cbr>􀝔 2 + 􀝕 2 = 􀝎 2 |\n| We now have an equation that involves 􀝔 and y only. r is a constant- notice that the equation we have found is the equation of acircle centred at (0,0) with radius r. |  |\n\nYou should be able to sketch a curve given its parametric equations-most of the time the easiest way to do this will be to convert to Cartesian form and then sketch, but if you are really struggling remember you can substitute values of t in and plot some points.  \nParabolas  \nYou have seen parabolas throughout mathematics, but you need to be able to identify and be confident working with the parametric form. The graph to the right is the graph of 􀝕2 = 4􀜽􀝔, where a is a positive constant.  \nAt first, this seems like a complicated way of defining a parabola, but this format allows the focus-directrix properties to appear much simpler.  \n• A parabola has Cartesian equation 􀝕 2 = 4􀜽􀝔, where a is a Positive constant. It has parametric equations  \n􀝔 = 􀜽􀝐 2, 􀝕 = 2􀜽􀝐, 􀝐 ∈ ℝ .  \nClearly this curve is symmetric about the x-axis, and a generic point can be defined as (􀝔, 􀝕)or (􀜽􀝐2, 2􀜽􀝐) .  \nAs well as being viewed as graphs of functions, parabolas, and all conic sections, can be defined geometrically. A parabola is the locus (set of all points), whose distance from the focus (a certain point) is equal to the distance from the directrix (a certain line) . For a parabola with cartesian equation 􀝕 2 = 4􀜽􀝔:  \n• The focus, denoted S, has co-ordinates (􀜽, 0)  \n• The directrix has equation 􀝔 + 􀜽 = 0  \n• The vertex is at the origin, (0,0) .  \nExample 2: Find the co-ordinates of the focus and the equation for the directrix of a parabola with equation 􀝕 2 = √80􀝔 .  \n\n| Put the equation into the form 􀝕 2 = 4􀜽􀝔 . | 􀝕 2 = 4√5􀝔, so, 􀜽 = √5 |\n| --- | --- |\n| Define the focus and directrix using the forms above. | Focus: (􀜽, 0) ⇒ (√5, 0)\u003Cbr>Directrix: 􀝔 + 􀜽 = 0 ⇒ 􀝔 + √5 = 0 |\n\nYou also need to be able to find the equation of a parabola from the focus and directrix:  \nExample 3: Find the equation of the parabola with focus (3,0) and directrix 􀝔 = −3  \n\n| Put the equation for the directrix in the form 􀝔 + 􀜽 = 0. | 􀝔 = −3 ⇒ 􀝔 + 3 = 0 |\n| --- | --- |\n| Define a. | Focus: (3,0), directrix: 􀝔 + 3 = 0, so 􀜽 = 3 |\n| Define the parabola-remember it is of the form 􀝕 2 ","cbCaidaKPNVcKpEb","https://ap.wps.com/l/cbCaidaKPNVcKpEb","pdf",332105,"English","# Conic sections overview\n## Parametric equations and conversion to Cartesian form\n# Parabolas\n## Focus-directrix properties and standard forms\n## Example problems and deriving focus/directrix\n# Rectangular hyperbolas\n## Cartesian and parametric forms\n# Tangents and normals\n## Gradient derivation using differentiation","[{\"question\":\"What is the relationship between conic sections and slicing cones?\",\"answer\":\"Conic sections are graphs formed by the intersection of a plane with a cone. Different slices produce shapes such as circles, ellipses, parabolas, and hyperbolas.\"},{\"question\":\"How do you convert parametric equations to a Cartesian equation?\",\"answer\":\"Eliminate the parameter between the x- and y-parameter equations, often using substitution. Watch for trigonometric identities that can simplify the elimination.\"},{\"question\":\"How are focus and directrix used to describe a parabola?\",\"answer\":\"A parabola is the locus of points whose distance from the focus equals the distance from the directrix. For a parabola in the form y^2 = 4ax, the focus is (a, 0) and the directrix is x = -a.\"}]","Conic Sections 1 - Conic Sections 1 Cheat Sheet | PDF",1791985239]