Grade 12

2012 - 2013

Term 2

Ahamed Mansour

Rashed Saeed

Omar Mohamed Fadel Mohamed

Task 1

Introduction :Write an introduction about each conic section, showing what they are used for.

1- Circle

· Introduction of circles:

A circle is a geometrical shape, and is not of much use in algebra, since the equation of a circle isn't a function. But you may need to work with circle equations in your algebra classes.

In algebraic terms, a circle is the set (or "locus") of points (x, y) at some fixed distance r from some fixed point (h, k). The value of r is called the "radius" of the circle, and the point (h, k) is called the "center" of the circle.

The "center-radius" form of the equation is:

(x – h)2 + (y – k)2 = r2

here the h and the k come from the center point (h, k) and the r2 comes from the radius value r. If the center is at the origin, so 
(h, k) = (0, 0), then the equation simplifies to x2 + y2 = r2.

The center-radius form of the circle equation comes directly from the Distance Formula and the definition of a circle. If the center of a circle is the point (h, k) and the radius is length r, then every point (x, y) on the circle is distance r from the point (h, k). Plugging this information into the Distance Formula (using r for the distance between the points and the center), we get:

· Used of circles:

1. Ring

2. Ferris Wheel

3. Clock

2-Ellipse

· Introduction of ellipse:

If you draw a line in the sand "through" these two sticks, from one end of the ellipse to the other, this will mark the "major" axis of the ellipse.

The points where the major axis touches the ellipse are the "vertices" of the ellipse. The point midway between the two sticks is the "center" of the ellipse.

The points where the minor axis touches the ellipse are the "co-vertices". A half-axis, from the center out to the ellipse, is called a "semi-major" or a "semi-minor" axis, depending on which axis you're taking half of.

The three letters are related by the equation b2 = a2 – c2 or, alternatively (depending on your book or instructor), by the equation b2 + c2 = a2.

For a wider-than-tall ellipse with center at (h, k), having vertices a units to either side of the center and foci c units to either side of the center, the ellipse equation is:

For a taller-than-wide ellipse with center at (h, k), having vertices a units above and below the center and foci c units above and below the center, the ellipse equation is:

· Used of ellipse:

1. Orbit of the earth

2. Race Track

3. Egg

3-Parabola

· Introduction of parabola:

In algebra, dealing with parabolas usually means graphing quadratics or finding the max/min points (that is, the vertices) of parabolas for quadratic word problems. In the context of conics, however, there are some additional considerations

The name "parabola" is derived from a New Latin term that means something similar to "compare" or "balance", and refers to the fact that the distance from the parabola to the focus is always equal .

The "general" form of a parabola's equation is the one you're used to, y = ax2 + bx + c — unless the quadratic is "sideways", in which case the equation will look something like x = ay2 + by + c. The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the x part is squared; for sideways (horizontal) parabolas, the y part is squared.

The "vertex" form of a parabola with its vertex at (h, k) is:

· Regular: y = a(x – h)2 + k

· 
Sideways: x = a(y – k)2 + h

· Used of parabola:

1. path of a thrown ball

2. McDonald's arches

3. Satellite dishes

4-Hyperbola

· Introduction of hyperbola:

The hyperbola is centered on a point (h, k), which is the "center" of the hyperbola. The point on each branch closest to the center is that branch's "vertex". The vertices are some fixed distance a from the center. The line going from one vertex, through the center, and ending at the other vertex is called the "transverse" axis. The "foci" of an hyperbola are "inside" each branch, and each focus is located some fixed distance c from the center.

When the transverse axis is horizontal (in other words, when the center, foci, and vertices line up side by side, parallel to the x-axis), then the a2 goes with the x part of the hyperbola's equation, and the y part is subtracted.

When the transverse axis is vertical (in other words, when the center, foci, and vertices line up above and below each other, parallel to the y-axis), then the a2 goes with the y part of the hyperbola's equation, and the x part is subtracted.

· Used of hyperbola:

1. Hourglass

2. Shine a flashlight

3. Cone shape wave that shoots out of the back of the plane

Gallery:

Go home, library, mall or other public places to find or take pictures for at least a picture or an item that represent each conic section then make a picture album

1- Circle

2-Ellipse

3-Parabola

4-Hyperbola

Task 2

Comprehensive comparison between conics Construct a table thatshows the similarities and differences between all types of conics? Showing the following points:

• Its definition

• Its equation

• Relation between its center and focus (foci)

• Other properties

• Graph it

Now classify the conics in the way you see it according to the similarities and differences.

Compare

1- Circle

2-Ellipse

3-Parabola

4-Hyperbola

Definition

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.

Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas.

is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus) and a line (the directrix).

hyperbola is a type of smooth curve, lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches.

Equation

Relation between its center and focus (foci)

the focus of a circle is its center.

If a line is perpendicular to the plane of the parabola and passes through the centre

Foci are points inside the hyperbola, when centre is not origin.When centre of hyperbola is (h,k)

Graph

Now classify the conics in the way you see it according to the similarities and differences.

The circle is a conic section where the plane is perpendicular to the axis of the cone. The special case of a point is where the vertex of the cone lies on the plane.

The ellipse is a conic section where the plane is not perpendicular to the axis, but its angle is less than one of the nappes. The special case of a point is where the vertex of the cone lies on the plane.

The parabola is a conic section where the plane is parallel to one of the nappes. The special case of two intersecting lines is where the vertex of the cone lies on the plane.

The hyperbole is a conic section where the angle of the plane is greater than on of the nappes. There are two sides to the hyperbole. The special case of two lines intersecting is where the vertex of the cone lies on the plane.

Task 3

A (0,3)

y=ax2+bx+c

3=a(0)2+b(0)+c

3=c

B (-2,7)

y=ax2+bx+c

7=a(-2)2+b(-2)+c

7=4a-2b+c

C (1,4)

y=ax2+bx+c

4=a(1)2+b(1)+c

4=a+b+c

Equation A subtract from equation B :-

3=c

-7=4a-2b+c

-4=-4a-b-+c-c

(-4=-4a-b) x1

4=4a+b

Equation A subtract from equation C :-

3=c

-4=a+b+c

-1=-a-b-+c-c

(-1=-a-b) x1

1=a+b

Find the value of A : -

b=a-1

4=4a+(a-1)

4=4a+a-1

4=5a-1

4+1=5a

5=5a

5a/5 = 5/5

a=1

Find the Value of B:-

b=a-1

b=1-1

b=0

so a equal to 1, b equal to 0 and C equal to 3.

Final equation:

y=ax2+bx+c

y=(1)x2+(0)x+c

y=x2+3

The final ; equation is y=x2+3

X+2y=2

2y=2-x

y=2/2-x/2

y=1-1/2x

x2+y2=25

x2+(1-1/2x)2=25

we expand the equation

x2+(1-1/2x)(1-1/2x)=25

x2+1-1/2x-1/2x+1/4x2=25

x2+1/4x2-1/2x-1/2x+1=25

5/4x2-x+1-25=0

5/4x2-x+24=0

(5x-24)(x+4)=0

x=24/5 , x=-4

then we subtract in y=1-1/2x to find y

y= 1-1/2x

y=1-1/2(24/5)

y=-7/5

when x =-4

y= 1-1/2x

y=1-1/2(-4)

y=3

The points are ( 24/5 , -7/6) and (-4,3)

( 24/5 , -7/6)

(-4,3)

Task 4

Vertex (0 ,20 )

Latus Rectum = 80

It is horizontal down word, so the equation is y=a(x-h)2+k

Vertex is ( h,k) = (0,20)

y=a(x-0)2+20

y=a(x)2+20

length of latus rectam = I1/aI units

80 = 1/a

a=-1/80

y=-1/80x2+20

axis of symmetry x=h , x=0.

-Focus (h, k+1/4a)

(0,20+1/4a)

(0,40)

-Directrix

y=k-1/4a

y=20-1/4(1/80)

y=0

The equation of the parabola is y=1/80x2+20

Find the center and values of a and b.

-The value of C is

c = 146/2=73

C=73

Difference of the distance from the comet to each body is 30.

a=30/2=15 million miles.

C2=a2+b2

732=152+b2

b2=5104.

The equation of the hyperbola

(x-h)2/a2 – (y-k)2/b2 =1

h=0 k=0

X2/152-y2/732=1

X2/225 –y2/5104 =1

Since the comet is farther from the sun ,it is located on the branch of the hyperbola near earth .

Website

http://omarmathematic.weebly.com/