--%>
Assignment Help - homework help

  Concepts of Conics and Parametric Equations

Conics:

The conics get their name from the fact that they can be made by passing a plane via a double-napped cone. There are mainly four conic sections, and three degenerate cases, though, in this class we are going to look at five degenerate cases which can be formed from the general second degree equation.

The general form of second degree equation is given by the Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Determining Conic Sections by Inspection:

To find out the conic section by inspection, complete any squares which are necessary, and hence the variables are on one side and the constant is on right hand side. Any squared variable beneath could be substituted by a quantity. That is, rather than x2 + y2 = 1, it may be (x-2)2 + y2 = 1

Circle:

x2 + y2 = 1

Both squared terms are existed, both are positive, both have similar coefficient. The right hand side is positive. When the right hand side is zero, then it is a point. If the right hand side is negative, then there is no graph.

Ellipse:

3x2 + 4y2 = 1

Both the squared terms are present, both are positive, however they have distinct coefficients. The right hand side should be positive. When the right hand side is zero, then it is a point. When the right hand side is negative, then there is no graph.

Hyperbola:

x2 - y2 = 1

Both the squared terms are present, however one is positive and the other is negative. The coefficients might or might not be similar, it does not matter. The right hand side is not zero. When the right hand side is zero, then it is intersecting lines.

Parabola:

x2 + y = 1

Both the variables are present, however one is squared and the other is linear.

Line:

x + y = 1

Neither variable is squared.

Point:

x2 + y2 = 0

The circle or ellipse with the right hand side is being zero.

No Graph:

x2 + y2 = -1

It is a circle or ellipse with the right hand side being negative.

Intersecting Lines:

x2 - y2 = 0

It is a hyperbola with the right hand side equivalent to zero.

Parallel Lines:

x2 = 1

One of the variable is squared and the other variable is missing. The right hand side should be positive. When the right hand side is zero, then this is a line (x2 = 0 therefore x = 0) and when the right hand side is negative (x2 = -1) then there is no graph.

Parabola:

2006_Parabola.jpg

The parabola is ‘the set of all points in a plane equidistant from a fixed point (that is, focus) and a fixed line (or directrix)’.

The distances to any point (x, y) on parabola from the focus (0, p) and the directrix y = -p, are equal to one other. This can be employed to develop the equation of the parabola.

When you take the definition of a parabola and work out algebra, you can build up the equation of a parabola. The short version is that, the standard form is x2 = 4py.

a) The starting point is a vertex at (h, k)

b) There is an axis of symmetry which contains the focus and vertex and is perpendicular to the directrix.

c) Move p units all along the axis of symmetry from the vertex to focus.

d) Move -p units all along the axis of symmetry from the vertex to directrix (which is a line).

e) The focus is in the curve.

The parabola consists of the property that any signal (light, sound and so on) entering the parabola parallel to the axis ofsymmetry will be reflected via the focus (this is why satellite dishes and such parabolic antennas which the detectives use to eavesdrop on the conversations work). As well, any signal originating at the focus will be reflected out parallel to axis of symmetry (this is why flash-lights work).

Circle:

The circle is ‘the set of all points in a plane equidistant from the fixed point (that is, center)’.

The standard form for a circle, with center at origin is x2 + y2 = r2, here r is the radius of the circle.

Ellipse:

The ellipse is ‘the set of all points in a plane in such a way that the sum of distances from the two fixed points (that is, foci) is constant’.

1479_Ellipse.jpg


The sum of distances to any point on the ellipse (x, y) from the two foci (c,0) and (-c,0) is constant. That constant will be equal to 2a.

When we let d1 and d2 bet the distances from foci to the point, then d1 + d2 = 2a.

We can use the definition to derive the equation of an ellipse; however the short form is shown below.

The ellipse is mainly a stretched circle. Start with the unit circle (that is, circle with radius of 1) centered at origin. Stretch the vertex from x = 1 to x = a and the point y = 1 to y = b. What you have done is multiplied each and every x by a andmultiplied each and every y by b.

In translation form, you symbolize that by x divided by a and y divided by b. Therefore, the equation of the circle modifies from x2 + y2 = 1 to (x/a)2 + (y/b)2 = 1 and that is the standard equation for an ellipse which is centered at origin.

a) Center is the beginning point at (h, k).

b) The major axis comprises the foci and the vertices.

c) Major axis length = 2a. This is as well the constant that the sum of distances should add to be.

d) Minor axis length = 2b.

e) Distance between the foci = 2c.

f) The foci are in the curve.

g) As the vertices are the furthermost away from center, a is the largest of three lengths, and the Pythagorean relationship is: a2 = b2 + c2.

Hyperbola:

347_Hyperbola.jpg

The hyperbola is ‘the set of all points in a plane in such a way that the difference of distances from the two fixed points (that is foci) is constant’.

The difference of distances to any point on the hyperbola (x, y) from two foci (c,0) and (-c,0) is constant. That constant will be equal to 2a.

When we let d1 and d2 bet the distances from the foci to point, then |d1 - d2| = 2a.

The absolute value is about the difference and hence it is always positive.

We can use that definition to derive the equation of hyperbola; however the short form is shown below.

The only difference in definition of a hyperbola and that of an ellipse is that, the hyperbola is the difference of distances from the foci that is constant and the ellipse is the sum of distances from the foci which is constant.

Rather than the equation being (x/a)2 + (y/b)2 = 1, the equation is (x/a)2 - (y/b)2 = 1.

The graphs, though, are much different.

a) Center is the beginning point at (h, k).

b) The Transverse axis includes the foci and the vertices.

c) Transverse axis length = 2a. This is as well the constant which is the difference of distances should be.

d) The conjugate axis length = 2b.

e) Distance between the foci = 2c.

f) The foci are in the curve.

g) As the foci are farthest away from the center, c is the largest of three lengths, and the Pythagorean relationship is: a2 + b2 = c2.

Standard Forms:

The table shown below summarizes the standard forms for the three main conic sections mainly based on the direction of main axis. For parabola, the axis is the ‘axis of symmetry’ and divides the parabola in half. For ellipse, it is termed as the ‘major axis’ and is the longer axis. For hyperbola, the axis is ‘transverse axis’ and goes among the vertices.

1421_Standard Forms of Conics.jpg

Latest technology based Algebra Online Tutoring Assistance

Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Algebra help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Algebra, project ideas and tutorials. We provide email based Algebra help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Algebra. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Algebra Homework help and assignment help services. They use their experience, as they have solved thousands of the Algebra assignments, which may help you to solve your complex issues of Algebra. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.