Chapter 10 : Quadratic Relations and Conic Sections
History of Conic Sections

History of Conic Sections
Apollonius of Perga (about 262-200 B.C.) was the last of the great mathematicians of the golden age of Greek mathematics. Apollonius, known as "the great geometer," arrived at the properties of the conic sections purely by geometry. His descriptions were so complete that he would have had little to learn about conic sections from our modern analytical geometry except for the improved modern notation. He did not, however, describe the properties of conic sections algebraically as we do today. It would take almost 2000 years before mathematicians would make great advances in the understanding of conic sections by combining both geometric and algebraic techniques. Apollonius defined the conic sections as sections of a cone standing on a circular base. The cone did not have to be a right cone, but could be slanted, or oblique. Apollonius noticed that all sections cut through such a cone parallel to its base were circles. He then extended the properties that he observed from these circles to ellipses and the other conic sections. He even solved the difficult problem of finding the shortest and longest distances from a given point to a conic section. These distances lie on lines called normals, which cut the curve of a conic section at right angles. Trying to read Apollonius' work, however, is not easy even with the best modern translation. Not only was his work very general in that it could be applied to many different situations involving conic sections, but it was also very long and rigorous, following the exacting principles of Euclidean proof. This is why the works of mathematicians like Hypatia, written several centuries later, were so important. Readers of Apollonius' Conics are better able to understand this text with the aid of these other texts that make the principles easier to understand. It is fortunate that scientists like Johannes Kepler, who...

...AN INTRODUCTION TO CONICSECTIONS
There exists a certain group of curves called ConicSections that are conceptually kin in several astonishing ways. Each member of this group has a certain shape, and can be classified appropriately: as either a circle, an ellipse, a parabola, or a hyperbola. The term "ConicSection" can be applied to any one of these curves, and the study of one curve is not essential to the study of another. However, their correlation to each other is one of the more intriguing coincidences of mathematics.
A CONICSECTION DEFINITION
Put simply, a conicsection is a shape generated when a cone intersects with a plane. There are four main types of conicsections: parabola, hyperbola, circle, and ellipse. The circle is sometimes categorized as a type of ellipse.
In mathematics, a conicsection (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a conic consists of those points whose distances to some point, called a focus, and...

...Chapter 13_Graphing the ConicSections
Ellipses
In this study guide we will focus on graphing ellipses but be sure to read and understand
the definition in your text.
Equation of an Ellipse (standard form)
Area of an Ellipse
( x − h) 2 ( y − k ) 2
+
=1
a2
b2
with a horizontal axis that measures 2a units, vertical axis
measures 2b units, and (h, k) is the center.
The long axis of an ellipse is called the major axis and the short
axis is called the minor axis. These axes terminate at points that
we will call vertices. The vertices along the horizontal axis will be
( h ± a, k ) and the vertices along the vertical axes will be ( h, k ± b) .
These points, along with the center, will provide us with a method
to sketch an ellipse given standard form.
A = π ab
Graph
( x − 5) 2 ( y − 8) 2
+
=1
9
25
First plot the center.
Then use a = 3 and
plot a point 3 units to
the left and 3 units to
the right of the
center.
Use standard form to
identify a, b, and the
center (h, k).
Next, use b = 5 and
plot a point 5 units up
and 5 units down
from the center.
Label at least 4
points on the ellipse.
In this example the major axis is the vertical axis and the minor axis is the horizontal
axis. The major axis measures 2b = 10 units in length and the minor axis measures
2a = 6 units in length. There are no x- and y- intercepts in this example.
Problems Solved!
13.4 - 1
Chapter 13_Graphing the...

...
1.
(a) Indications
-The stenopaic slit refraction is useful for confirming the results of other refraction techniques for patients with irregular astigmatism or reduced visual acuity.
- It is helpful for patients who have difficulty understanding the complex instructions associated with other subjective techniques.
-It is important to note that, like the pinhole, the stenopaic slit may be used diagnostically to determine a patient's potential visual acuity.
-The astigmatism present in the patient’s old spectacles should be considered
-The small amount of cylindrical power is of little consequences, in that the subjective end point can quickly be rechecked after the stenopaic slit is removed
(b) Techniques
Axis determination
i. remove cylinder power from retinoscopy and
ii. identify BVS or MPTMV, VA
iii. fogged the patient (F=1/2 CYL + 0.50 DS) and watch the acuity chart
iv. put the stenopaic slit at any position
v. the slit is rotated until acuity is maximized. The slit now lies along the minus cylinder axis.
Spherical power
vi. with the slit in this position, the fog is reduced to best acuity.
vii. the lens in place is the sphere power of the patient’s lens formula (pt’s final sphere)
viii. the slit is rotated 90 degrees. This will fog the patient again
viiii. the fog is again reduced to best acuity. The algebraic difference between the power of the lens in place at the end of this operation and the lens power in place at the end of step one is the minus cylinder...

...Paper - I
1. Sources: Archaeological sources:Exploration, excavation, epigraphy, numismatics, monuments Literary sources: Indigenous: Primary and secondary; poetry, scientific literature, literature, literature in regional languages, religious literature. Foreign accounts: Greek, Chinese and Arab writers.
2. Pre-history and Proto-history: Geographical factors; hunting and gathering (paleolithic and mesolithic); Beginning of agriculture (neolithic and chalcolithic).
3. Indus Valley Civilization: Origin, date, extent, characteristics, decline, survival and significance, art and architecture.
4. Megalithic Cultures: Distribution of pastoral and farming cultures outside the Indus, Development of community life, Settlements, Development of agriculture, Crafts, Pottery, and Iron industry.
5. Aryans and Vedic Period: Expansions of Aryans in India. Vedic Period: Religious and philosophic literature; Transformation from Rig Vedic period to the later Vedic period; Political, social and economical life; Significance of the Vedic Age; Evolution of Monarchy and Varna system.
6. Period of Mahajanapadas: Formation of States (Mahajanapada): Republics and monarchies; Rise of urban centres; Trade routes; Economic growth; Introduction of coinage; Spread of Jainism and Buddhism; Rise of Magadha and Nandas. Iranian and Macedonian invasions and their impact.
7. Mauryan Empire: Foundation of the Mauryan Empire, Chandragupta, Kautilya and Arthashastra; Ashoka;...

...separates from the choroid after a retinal
tear develops.
Retinal detachment is a serious eye condition. If it is not treated, it can lead to
blindness. Each year, 30,000 people in the United States are diagnosed with retinal
detachment.
There are clear warning signs that a person is developing a retinal tear or
detachment. When diagnosed early, most retinal problems are treatable. With
treatment, retinal problems usually do not affect vision very much.
This reference summary explains what retinal tears and detachments are. It discusses
their symptoms, causes, diagnosis and
Lens
treatment options.
Cornea
Anatomy
It is important to recognize the parts of the
eye before learning about retinal tears and
detachments. This section reviews the
anatomy of the eye.
Light hits the cornea of the eye first. The
cornea is the transparent covering on the
front of the eye.
Iris
Vitreous
Macula
Retina
Next, light travels to the back part of the eye through the pupil. The pupil is the
opening in the center of the iris, the colored part of the eye.
This document is a summary of what appears on screen in X-Plain™. It is for informational purposes and is not intended to be a substitute for the advice
of a doctor or healthcare professional or a recommendation for any particular treatment plan. Like any printed material, it may become out of date over
time. It is important that you rely on the advice of a doctor or a healthcare...

...A conic or conicsection is a plane curve that can be obtained by intersecting a cone with a plane that does not go through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and the plane. If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse. If one line of the cone is parallel to the plane, the intersection is an open curve whose two ends are asymptotically parallel; this is called a parabola. Finally, there may be two lines in the cone parallel to the plane; the curve in this case has two open pieces, and is a hyperbola.
In mathematics, parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of functions from items such as R. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.
Converting a set of parametric equations to a single equation involves solving one of the equations usually the simplest of the two for the parameter. Then the solution of the...

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...
Today one of the most cherished ideologies of America is the fact that everyone is and should be created equal. With this cherished ideology bringing a sense of pride and diversity to America we must keep in mind that this cherished ideology did not always exist. Since 1865 various individuals and groups have not been able to receive and express their rights to full equal status in the United States. These different individuals and groups have seemingly fought for their rights in equality and have become pioneers in the fight for evolution for equality.
In 1865 African Americans in the United States under the 13th amendment were freed from the terrible burden of slavery. Through the 14th amendment they were given the right to citizenship and the right to equal protection. The 15th amendment gave them the right to vote regardless of their skin color race or any other type of servitude. These amendments were meant to be enforced and make a serious change in the everyday life of the average American.
With these amendments passing in 1865 they were meant to make a serious change towards the evolution of equality. These changes did not seem to happen right away and African Americans were still not being treated with equality. The average African American at this time were being denied there newly given rights every day making life extremely hard to stay...