435759541-Conic-Sections-Ellipse-Final.pptx
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The document defines and describes key properties of ellipses, including that an ellipse is the set of all points where the sum of distances from two fixed foci is a constant. It provides the standard equation forms for ellipses with horizontal and vertical major axes when the center is at (0,0), and then extends this to ellipses with an arbitrary center (h,k). Key properties defined include the lengths of the semi-major and semi-minor axes, the distance between the center and foci, and the Pythagorean relationship between these values. Examples are given to assess understanding of writing the standard equation for an ellipse given properties like the vertices and foci.
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https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdAdvanced Java[Extra Concepts, Not Difficult].docx
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ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
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Natural birth techniques - Mrs.Akanksha Trivedi Rama University
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435759541-Conic-Sections-Ellipse-Final.pptx
- 1. Ellipse with V(h,k) Application of Ellipse
- 2. Ellipse The ellipse is the set of all points on the plane for which the sum of distances from two fixed points (foci) is a positive constant.
- 3. Parts of an Ellipse: Semi-axis - ½ the length of axis Center - midpoint of the segment joining the foci Vertices - endpoints of the major axis. The points on the ellipse which lie on the line containing the foci Major axis – The line segment joining the vertices, longer axis, contains foci Co- Vertices – the points on the ellipse which are on the perpendicular bisector of the major axis Minor axis – The line segment joining the co-vertices, shorter axis Latus Rectum - The line segment passing through the focus and perpendicular to the major axis Foci - two given fixed points on the major axis Center Focus Focus
- 4. Equation
- 5. Ellipse with horizontal major axis: • Center is (0, 0). • Length of major axis is 2a. • Length of minor axis is 2b. • Distance between center and either focus is c with c2 = a2– b2, a > b > 0.
- 6. Ellipse with vertical major axis: • Center is (0, 0). • Length of major axis is 2a. • Length of minor axis is 2b. • Distance between center and either focus is c with c2 = a2– b2, a > b > 0. • LR = 2b2/a • e = c/a • d = a/e = a2/c
- 7. Problem:
- 9. Ellipse with horizontal major axis: • Center is (h, k). • Length of major axis is 2a. • Length of minor axis is 2b. • Distance between center • and either focus is c with c2 = a2– b2, a > b > 0.
- 10. Ellipse with vertical major axis: • Center is (h, k). • Length of major axis is 2a. • Length of minor axis is 2b. • Distance between center and either focus is c with c2 = a2– b2, a > b > 0.
- 11. Pre- Calculus Ellipses with Center (h, k) Standard Equation Focal Axis y = k x = h Foci (h c, k) (h, k c) Vertices (h a, k) (h, k a) Semimajor Axis a a Semiminor Axis b b Pythagorean Relation a2 = b2 + c2 a2 = b2 + c2 2 2 2 2 (x h) (y k) 1 a b 2 2 2 2 (y k) (x h) 1 a b
- 12. ASSESSMENT 1. What is the standard form of the ellipse that has vertices at (0, ±3) and foci at ( 0, ±2)? 2. What is the standard form of equation of the ellipse that has vertices (-2, -8) and (-2, 2) and foci (-2, -7) and (-2, 1).?