The parabola has an electromagnetic signal reflection property. Four signals are shown in green and blue. These signals are shown with arrows on both ends to indicate the focus either collects the signals (coming in) or the focus generates the signals and they leave in parallel from the parabola. The inside of the parabola can be a mirror (for light) or another material (for non visible electromagnetic waves.) As light enters parallel to axis of symmetry it will strike the parabola and reflect toward the focus. You can see heavy black line segments drawn on the parabola on the lines tangent to the parabola at the points of incidence. Two angles are formed between each of these segments and the light striking and bouncing off; each pair of angles are equal and depend upon the location the light hits the parabola. Imagine the focus is a light bulb and the parabola a mirror. The light bulb emits light in all directions. All the light that strikes the parabola will leave parallel to the axis of symmetry. Spot lights make use of this property. Of course a a parabolic mirror is 3-dimensional. Imagine rotating the parabola about its axis of symmetry and you will get a shape you'll recognize as the headlight of your car.

Light emitted from the focus leaves the parabolic mirror in parallel paths, shown below. Headlights, spotlights, etc., have the shape of a parabola to increase the intensity of the light and direct the light. The ellipse has similar reflective properties. Below you see three lines, blue, green and red. These lines start from either focus and 'bounce' off the ellipse toward the other focus. As with the parabola, the angles between each signal and its tangent line segment (dark black) are equal.

The parabola also amplifies any signal entering it directing it to the focus. Satellite dishes use this property as do dishes used at astronomical observatories.

...Conics are surprisingly easy! There are four types of conicsections, circles, parabolas, ellipses, and hyperbolas. The first type of conic, and easiest to spot and solve, is the circle. The standard form for the circle is (x-h)^2 + (y-k)^2 = r^2. The x-axis and y-axis radius are the same, which makes sense because it is a circle, and from
In order to graph an ellipse in standard form, the center is first plotted (the (h, k)). Then, the x-radius is plotted on both sides of the center, and the y-radius is plotted both up and down. Finally, you connect the dots in an oval shape. Finally, the foci can be calculated in an ellipse. The foci is found in the following formula, a^2 b^2 = c^2. A is the radius of the major axis and b is the radius of the minor axis. Once this is found, plot the points along the major axis starting from the center and counting c amount both directions.
In order to determine if an equation is an ellipse, the following three criteria must be met. There must be an x^2 and a y^2 just like in a circle. However, the coefficients of the x^2 and y^2 must be different. Finally, the signs must be the same. For example, equation 4 is an ellipse. 49x^2 + 25y^2 +294x 50y 759 = 0 has an x^2 and a y^2. It also has different coefficients in front of them, and finally, both have the same sign! There you have it, an ellipse!HyperbolasBoy, now it is starting to get tough! But dont worry, hyperbolas are not...

...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...

...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...

...Chapter 10 : Quadratic Relations and ConicSections
History of ConicSections
History of ConicSections
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 conicsections purely by geometry. His descriptions were so complete that he would have had little to learn about conicsections from our modern analytical geometry except for the improved modern notation. He did not, however, describe the properties of conicsections algebraically as we do today. It would take almost 2000 years before mathematicians would make great advances in the understanding of conicsections by combining both geometric and algebraic techniques.
Apollonius defined the conicsections 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 conicsections. He even solved the difficult problem of finding the shortest...

...see that everyone just called ellipses circles because they were both round. After they circle was defined, the ellipse did not fit in the description, and scientists began to study them. It wasn’t until the early 1600s when the focus was discovered by Kepler. Scientists first started thinking about the ellipse while studying the orbit shape of the planets. After years of research, they decided to make a new conic shape instead of calling all of the ellipses circles. They knew the seasons which helped them consider that the earth is not revolving in a circle, but rather in an oval.
The length of an ellipse wasn’t discovered until 1914 by Ramanujan. According to Kepler's First Law of Planetary Motion, the orbit of each planet is an ellipse, with one focus of that ellipse at the center of the Sun. Newton's reconsideration of this law states that the orbit of each planet is a conicsection, with one focus of that conicsection at the center of the Sun. To properly understand planetary orbits, we therefore need some understanding of ellipses in particular and conicsections in general.
Ellipses are used in our everyday lives. It is important to know where they came from and how far they date back (Even though they have been here since the earth was created).
Since you see them everywhere, why not try and learn more about them?
Source:...

...cbse.entrancei.com
APPLICATION OF INTEGRALS - NCERT SOLUTIONS
Question 1:
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the xaxis.
ANSWER
The area of the region bounded by the curve, y2 = x, the lines, x = 1 and x = 4, and the x-axis
is the area ABCD.
Question 2:
Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first
quadrant.
ANSWER
1
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cbse.entrancei.com
The area of the region bounded by the curve, y2 = 9x, x = 2, and x = 4, and the x-axis is the
area ABCD.
Question 3:
Find the area of the region bounded by x2 = 4y, y = 2, y = 4 and the y-axis in the first
quadrant.
ANSWER
2
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cbse.entrancei.com
The area of the region bounded by the curve, x2 = 4y, y = 2, and y = 4, and the y-axis is the
area ABCD.
Question 4:
Find the area of the region bounded by the ellipse
ANSWER
The given equation of the ellipse,
, can be represented as
Question 5:
Find the area of the region bounded by the ellipse
ANSWER
The given equation of the ellipse can be represented as
3
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cbse.entrancei.com
It can be observed that the ellipse is symmetrical about x-axis and y-axis.
∴ Area bounded by ellipse = 4 × Area OAB
Therefore, area bounded by the ellipse =
Question 6:
Find the area of the region in the first quadrant enclosed by x-axis, line
and the
circle
ANSWER...

...
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...

...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...