Before we embark on solving the problem, let us first explore the definition of ellipse.

[pic]
An ellipse is a curve that is the locus of all points in the plane the sum of whose distances [pic] and [pic] from two fixed points [pic] and [pic] (the foci) separated by a distance of [pic] is a given positive constant [pic]

[pic]
While [pic] is called the major axis, [pic] is the semi major axis, which is exactly half the distance across the ellipse. Similarly, the corresponding parameter [pic] is known as the semi minor axis.

Parallels drawn from the formula for the area of circle ([pic]) and formula for the area of an ellipse (A = [pic]ab)

Formula for the area of a circle:[pic] where [pic] is the area, and [pic] is the radius. In the case of a circle, radius a represents the semi major axis while radius b represents the semi minor axis. One can thus find the area of the circle through the formula A = [pic]ab, where a is equal to b. Hence circle, in actual fact, is a unique case of ellipse.

Proving that the area of an ellipse is πab

Procedures to take (Theory)

1. We have to let an ellipse lie along the x-axis and find the equation of the ellipse curve. 2. Upon finding the equation of the ellipse curve, we have to change the subject of the equation to y. (otherwise, we will have to do integration with respect to y if the subject of the equation is x.) 3. Next, applying what we have learnt, we can find the area bounded by the ellipse curve through definite integration between the 2 limits.

Calculations (Practical)
(Working steps are adapted from http://mathworld.wolfram.com/Ellipse.html)

Let an ellipse lie along the x-axis and find the equation of the figure where [pic]and [pic]are at [pic]and [pic]. a) Form an equation.
[pic]
b) Bring the second term to the right side and square both sides. [pic]
c)...

...There are four types of conic sections, 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 much...

...plane curve such that the sums of the distances of each point in its periphery from two fixed points, the foci, are equal. It is a conic section (see conic) formed by the intersection of a cone by a plane which cuts obliquely the axis and the opposite sides of the cone. The ellipse is a conic which does not extend to infinity, and whose intersections with the line at infinity are imaginary. Every ellipse has a center, which is a point such that it bisects every chord passing through it. Such chords are called diameters of the ellipse. A pair of conjugate diameters bisect, each of them, all chords parallel to the other. The longest diameter is called the transverse axis, also the latus transversum; it passes through the foci. The shortest diameter is called the conjugate axis. The extremities of the transverse axis are called the vertices. (See conic, eccentricity, angle.) An ellipse may also be regarded as a flattened circle—that is, as a circle all the chords of which parallel to a given chord have been shortened in a fixed ratio by cutting off equal lengths from the two extremities. The two lines from the foci to any point of an ellipse make equal angles with the tangent at that point. To construct an ellipse, assume any line whatever, AB, to be what is called the latus rectum. At its extremity erect the perpendicular AD of any length, called the latus transversum (transverse axis)....

...Conic Sections
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....

...
Ellipse Construction, ContinuedParallelogramThe parallelogram method of constructing ellipses inscribes the ellipse withinellipsesa parallelogram. You may use conjugate diameters or the major and minoraxes to formulate the parallelogram so long as the sides of the parallelogramare parallel to the diameters or axes.
step1ActionGiven the major and minor axes or the conjugate diameters AB andCD, draw a rectangle or parallelogram . Make sure all sides are parallel to their respective sides.
2Divide the distance between AO and AJ into the same number ofequal parts.
3Starting at the ends of the minor axis CD, lightly draw straight linesthrough each point. The lines intersect forming the circumferenceof the ellipse.
4Lightly sketch the outline of the ellipse. Darken the outline usingfrench curves.
To draw an ellipse by the parallelogram method, use this table;
Focus-Directrix orEccentricityMethod
Given :the distance offocusfrom the directrix and eccentricity
Example : Draw an ellipse if the distance of focus from the directrix is 70
mm and the eccentricity is 3/4.
1. Draw the directrix AB and
axis CC’
2. Mark F on CC’ such that
CF= 70 mm.
3. Divide CF into 7 equal
parts and mark V at the
3
parts and mark V at the
fourth division from C.
Now, e = FV/ CV = 3/4.
4. At V, erect a perpendicular
VB = VF. Join CB. Through
F, draw a line at 45°to meet...

... SECTION A (40 Marks)
Attempt all questions from this Section
Question 1.
a) What number must be subtracted from 2x3 – 5x2 + 5x so that the resulting polynomial has a factor 2x – 3 ? [3]
b) D, E, F are mid points of the sides BC, CA and AB respectively of a Δ ABC. Find the ratio of the areas of Δ DEF and Δ ABC. [3]
c) A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5% per annum, find the sum borrowed. [4]
Question 2.
a) The y-axis is a line of symmetry for the figure ACBD where A, B have co-ordinates (3, 6), (– 3, 4) respectively. (i) Find the co-ordinates of C and D. (ii) Name the figure ACBD and find its area. [3]
b) PAQ is a tangent at A to the circumcircle of Δ ABC such that PAQ is parallel to BC, prove that ABC is an isosceles triangle. [3]
c) A rectangular piece of paper 30 cm long and 21 cm wide is taken. Find the area of the biggest circle that can be cut out from this paper. Also find the area of the paper left after cutting out the circle. [Take π = 22/7] [4]
Question 3.
a) Construct a 2 × 2 matrix whose elements aij are given by aij = i + j. [3]
b) The point P (– 4, – 5) on reflection in y-axis is mapped on P’. The point P’ on reflection...

...Surface Area Formulas
In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.
Cube | Rectangular Prism | Prism | Sphere | Cylinder | Units
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
|Surface Area of a Cube = 6 a 2 |
[pic](a is the length of the side of each edge of the cube)
In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.
|Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac |
[pic](a, b, and c are the lengths of the 3 sides)
In words, the surface area of a rectangular prism is the are of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths...

...Explain each of the areas of learning and development and how these are interdependent.
Personal social and emotional has three aspects, making relationships, managing feelings and behaviour and self-confidence and self-awareness. This area is all about the child’s relationships with other people and themselves. Children need to develop relationship with the people around the for example the children they play with and come into contact with. The staff that work in that room and their family members. You recognise the relationship people have by observing them and if it’s you and a child you know if that child is close to you. Managing feeling and behaviour is all about the feeling children get and what they know about them and can they recognise them within other people. And the behaviour side of it is knowing what to do and what not to do and can tell if there is any harm coming towards anyone. They can also recognise the behaviours in other children. Self-confidence and self-awareness is about the child knowing when they have done something new the can say look I have done this and are happy about it. Generally aware of themselves and what they are doing.
Physical development has two areas of development in one being moving and handling. This is every physical movement that the child does so from drawing with a pencil using a three finger grip to jumping around outside and climbing the pirate ship. The other...

...
(6.03)
61) What is the area of a square that has a side length of:
a) 9 units
b) 11 units
Area of a square: = (Side length)2
a) A = (9)2 b) A = (11)2
= 81 u2 = 121 u2
62) A rectangle has a length of 8 in and a width of 5 in. Find the area.
Area of a rectangle: = (L)(W)
A = (L)(W)
=(8 in)(5 in)
= 40 in2
63) Find the length of a rectangle if the width is 8 cm and the perimeter 40 cm.
Perimeter: = 2(L + W)
40 = 2(L + 8)
2 2
20 = L + 8
- 8 - 8
12 = L
L = 12 cm
64) The diagonals of a square measure 12 cm. Find the area of the square. (Show two methods)
[pic]
Method 1: Method 2:
Using Pythagorean Theorem Introducing Diagonal Formula
a2 + a2 = 122 Area = (d1)(d2)
2(a2) = 144 2
2 2 = (12)(12) 6
a2 = 72 2 1
Area = (a)(a) Area = 72 cm2
= a2
Area = 72 cm2
65) Prove the following: If the...