The document discusses key concepts in Cartesian geometry including:
1) The Cartesian plane consists of two perpendicular number lines that intersect at the origin point and is used to describe the position of points using coordinates.
2) Key parts of the Cartesian plane include the x- and y- axes and formulas are provided for calculating the distance between two points and finding the midpoint of a line segment.
3) Equations and graphical representations are provided for circles, parabolas, ellipses, and other conic sections along with examples of using the equations.
(Distancia. Punto Medio. Ecuaciones y trazado de circunferencias, Parábolas, elipses, hipérbola. Representar gráficamente las ecuaciones de las cónicas).
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
(Distancia. Punto Medio. Ecuaciones y trazado de circunferencias, Parábolas, elipses, hipérbola. Representar gráficamente las ecuaciones de las cónicas).
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Instructions for Submissions thorugh G- Classroom.pptx
Plano numerico
1. PLANO
NUMERICO
Republica Bolivariana De Venezuela
Ministerio del Poder Popular para la Educación Universitaria
Universidad Politécnica Territorial Del Estado Lara Andrés Eloy Blanco
Barquisimeto Estado Lara
Barquisimeto, Febrero del 2021
Participante:
Michelle González
28.732.056
PNFHSL 0103
Turno mañana
UC Matemáticas
2. Plano Cartesiano
Se conoce como plano cartesiano, coordenadas cartesianas o sistema cartesiano, a dos rectas
numéricas perpendiculares, una horizontal y otra vertical, que se cortan en un punto llamado
origen o punto cero, la finalidad del plano cartesiano es describir la posición o ubicación de un
punto en el plano, la cual está representada por el sistema de coordenadas. El plano cartesiano
también sirve para analizar matemáticamente figuras geométricas como la parábola, la hipérbole,
la línea, la circunferencia y la elipse, las cuales forman parte de la geometría analítica.
.
Partes del plano cartesiano
2
3. Distancia
La distancia entre dos puntos sobre un plano es simplemente la distancia mínima que hay entre ambas
posiciones, las cuales vienen determinadas por las sus coordenadas en el eje de las X y en el eje de las Y.
La distancia mínima es sinónimo del camino más corto que separa a ambas singularidades.
Formula.
Sean dos puntos sobre el plano
cartesiano,𝑃1(𝑥1, 𝑦1) y 𝑃2(𝑥2, 𝑦2). La
distancia que hay entre ellos viene dada por
la siguiente expresión:
𝑑 𝑃1, 𝑃2 = (𝑋𝑥 − 𝑥1)2 + (𝑦2 − 𝑦1)2
Ejemplo.
Halla la distancia en el plano entre dos puntos cuyas coordenadas
cartesianas son 𝑃1(3,2) y 𝑃2 5,1
Simplemente tenemos que introducir de forma adecuada los datos del
enunciado, operar y listo:
✔ 𝑃1(𝑥1, 𝑦1) viene dado por 𝑃1 3,1
✔ 𝑃2(𝑥2, 𝑦2 ) es 𝑃2(5,6)
Entonces: 𝑑(𝑃1, 𝑃2 = 𝑥2 − 𝑥1
2 + (𝑦2, 𝑦1)2 =
(5 − 3)2+(6 − 1)2
Operando: 𝑑 𝑃1, 𝑃2 = (2)2+(5)2= 4 + 25 = 29 ≈ 5.385 3
4. Punto Medio
Formula
Dado un segmento, cuyos extremos
tienen por coordenadas:
𝐴 = (𝑥1, 𝑦1) y 𝐵 = (𝑥2, 𝑦2)
El punto medio, 𝑃𝑚, tendrá por
coordenadas
𝑃𝑚=
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
Ejemplo
Halla las coordenadas del punto medio del segmento AB donde los extremos son:
a) 𝐴 3,9 𝑦 𝐵 −1,5
b) 𝐴 7,3 𝑦 𝐵 −1,5
✔ Primer caso: 𝑃𝑚
3−1
2
,
9+5
2
= 𝑃
𝑚(1,7)
El punto medio es 𝑃𝑚(1,7)
✔ Segundo caso: 𝑃𝑚
7−1
2
,
3+5
2
= 𝑃𝑚(3,4)
El punto medio es 𝑃𝑚(3,4) 4
En el plano cartesiano, encontrar el punto medio significa encontrar las coordenadas de un punto 𝑃𝑚 en el
segmento que une a 𝑃1 con 𝑃2, tal que la distancia entre 𝑃1 y 𝑃𝑚 es igual a la distancia entre 𝑃2 y 𝑃𝑚, es
decir, 𝑃𝑚 es un punto equidistante a 𝑃1 y 𝑃2 y que se encuentra sobre el segmento que une 𝑃1 con 𝑃2.
5. Ecuaciones y trazado de
circunferencia
✔ Ecuaciones: La ecuación de una circunferencia centrada en el punto
C(a,b) y con radio r se puede escribir de la siguientes formas:
1 (𝑥 − 𝑎)2 + 𝑦 − 𝑏 2 = 𝑟
2 𝑥2
+ 𝑦2
+ 𝑚𝑥 + 𝑛𝑦 + 𝑝 = 0
Donde: 1) 𝑚 = −2𝑎 2) 𝑛 = −2𝑏 3) 𝑝 = 𝑎2
+ 𝑏2
− 𝑟2
✔ Trazado: Para este uso de la demostración
la ecuación siguiente para un circunferencia:
(x + 1)2 + (y - 2)2 = (1.5)2.
Ejemplo:
Desde 𝑥 − ℎ = 𝑥 + 1, ℎ = −1
Desde 𝑦 − 𝑘 = 𝑦 − 2, 𝑘 = 2.
El centro de la circunferencia esta Tan en el punto
−1,2
Trace el punto (−1,2)
5
Una circunferencia es el conjunto de todos los puntos de un plano que equidistan de otro punto fijo y coplanario llamado centro. A
la distancia entre cualquiera de sus puntos y el centro se le denomina radio.
Ejemplo
Determina la existencia de la ecuación de la circunferencia
𝑥2
+ 𝑦3
− 2𝑥 + 2𝑦 + 5 = 0
Podemos extraer que 𝑚 = −2, 𝑛 = 2 y p = 5
De igual forma, a partir de estos coeficiente podemos determinar que el centro
C(a,b) de la circunferencia es 𝑎 =
𝑚
−2
= 1, 𝑏 =
𝑛
−2
= 1.Por lo tanto, la
circunferencia se encuentra centrada en C(1,-1)
Evaluando la expresión 𝑎2
+ 𝑏2
− 𝑝 Obtenemos que:
𝑎2
+ 𝑏2
− 𝑝 = 1 2
+ −1 2
− 5 = −3 < 0
Dado que el valor obtenido es inferior a cero la circunferencia del enunciado no
existe
6. Parábolas
6
En el Plano Cartesiano una parábola puede tener su vértice en cualquier par de coordenadas y puede estar orientada hacia arriba,
hacia abajo o hacia la izquierda o la derecha. Esta ecuación tiene leves variaciones según sea la orientación de la parábola (hacia
donde se abre).
Primera posibilidad: La que ya vimos, cuando
la parábola se abre hacia la derecha
(sentido positivo) en e l eje de las abscisas “ X”
𝑦2
= 4𝑝𝑥
Segunda posibilidad: Cuando la parábola se abre hacia
la izquierda (sentido negativo) del eje de las abscisas
“ X”.
𝑦2
= 4𝑝𝑥 (𝑐𝑜𝑛 𝑠𝑖𝑔𝑛𝑜 𝑚𝑒𝑛𝑜𝑠 𝑓𝑖𝑛𝑎𝑙)
Tercera posibilidad: Cuando la parábola se abre
hacia arriba (sentido positivo) en el eje de
las ordenadas “ Y” .
𝑦2
= 4𝑝𝑥 (𝑐𝑜𝑛 𝑠𝑖𝑔𝑛𝑜 𝑚𝑒𝑛𝑜𝑠 𝑓𝑖𝑛𝑎𝑙)
Cuarta posibilidad: Cuando la parábola se abre hacia
abajo (sentido negativo) en el eje de las
ordenadas “Y”.
𝑥2
= 4𝑝𝑦 (𝑐𝑜𝑛 𝑠𝑖𝑔𝑛𝑜 𝑚𝑒𝑛𝑜𝑠 𝑓𝑖𝑛𝑎𝑙)
Hallar la ecuación de la parábola de directriz x=4x=4 y foco F(–
2,0)F(–2,0).
• El valor absoluto de cc es la distancia del vértice al foco. 𝑐 =
𝑑(𝑉, 𝐹)
• El vértice esta sobre el eje focal y a la misma distancia del foco
que de la directriz 𝑉 =
−2+3
2
, 0 = (10, )
• Eje focal: eje x . Como el eje es horizontal la ecuación tiene
forma:
(𝑦 − 𝛽)2 = 4𝑐(𝑥 − 𝑎)
(𝑦 − 0)2 = 4𝑐(𝑥 − 1)
• Falta calcular el valor absoluto de c.
𝑐 = 𝑑 𝐹, 𝑉 = 3
• Como el foco esta a la izquierda del vértice entonces c=-3
• Entonces nos queda: 𝑦2 = −12(𝑥 − 1)
Ejemplo
7. Elipses
✔ Ecuación de eje mayor horizontal centrada a un punto
cualquiera P(𝒙𝟎, 𝒚𝟎).
Esta ecuación viene dada por :
(𝑥−𝑥0)2
𝑎2 +
(𝑦−𝑦0)2
𝑏2 = 1
Donde:
• 𝑥0, 𝑦0 : Coordenadas x e y del centro del elipse
• A: Semieje de abscisas
• B: Semieje de ordenadas. En nuestro caso debe cumplirse que
𝑏 ≤ 𝑎
✔ Ejemplo:
Determina la ecuación de la elipse horizontal centrada en el origen
cuyo eje mayor horizontal mide 10 y su distancia focal mide 6.
• Dado que sabemos que el eje mayor es 2. 𝑎) es 10:
2𝑎 = 10 𝑎 = 5
• Y que la distancia focal (2. 𝑐) mide 6:
2𝑐 = 6 𝑐 = 3
• Partiendo de estos datos, podemos calcular la longitud del semieje menor
(b) por medio de la siguiente ecuación:
𝑏2
= 𝑎2
− 𝑐2
𝑏2
= 52
− 32
𝑏 = 25 − 9 𝑏 = ±4
• Dado que no puede existir una longitud negativa nos quedaremos con que
b=4. Utilizando ahora la formula de la ecuación de una elipse mayor
horizontal situada en el punto P(0,0) o lo que es lo mismo 𝑥𝑐 = 0 e 𝑦0 =
0.
(𝑥 − 𝑥0)2
𝑎2
+
(𝑦 − 𝑦0)2
𝑏2
= 1
(𝑥 − 0)2
52
+
(𝑦 − 0)2
42
= 1
𝑥2
25
+
𝑦2
16
= 1
7
La elipse se define como una línea
curva cerrada tal que la suma de las
distancias a dos puntos fijos, F y F' ,
llamados focos, es constante. Ten en
cuenta que para cualquier punto de la
elipse siempre se cumple que: 𝑑 𝑃, 𝐹 +
𝑑 𝑃, 𝐹′ = 2. 𝑎
Donde 𝑑 𝑃, 𝐹 + 𝑑(𝑃, 𝐹‘) es la distancia
de un punto genérico 𝑃 al foco 𝐹‘
respectivamente
8. Representar gráficamente las
ecuaciones de las cónicas
✔ Ejemplo
1 + 3𝑥 + 2𝑦 + 𝑥2
− 4𝑥𝑦 + 7𝑦2
= 0
1 𝑥 𝑦
1
3
2
1
3
2
1 − 2
1 − 2 7
1
𝑥
𝑦
= 0.
8
Una cónica es el lugar geométrico de los puntos del plano (x,y) que satisfacen una ecuación completa de
segundo grado:
𝐵𝑥2
+ 𝐶𝑦2
+ 𝐷𝑥𝑦 + 𝐹𝑥 + 𝐺𝑦 + 𝐻 = 0
La ecuación de una cónica se puede escribir en forma matricial como;
1 𝑥 𝑦
𝑎00 𝑎01 𝑎02
𝑎10 𝑎11 𝑎12
𝑎20 𝑎21 𝑎22
1
𝑥
𝑦
= 0.
