Equation of a Circle
Equation of a circle with centre (a,b) and radius r.



PR = x - a, PQ = r

QR = y - b

Since ΔPQR is the right angle triangle, we have: PQ2 = PR2 + QR2

r2 = (x - a)2 + (y - b)2

if the center of the circle is the origin (0, 0), the equation becomes x2 + y2 = r2

General Equation of a Circle

From (x - a)2 + (y - b)2 = r2

r2 - 2ax + a2 + y2 - 2by + b2 - r2 = 0

x2 + y2 - 2ax - 2by + a2 + b2 - r2 = 0

The above equation can be written as x2 + y2 + 2gx + 2fy + c = 0

Where a = -g, b = -f, c = -a2 + b2 - r2

Hence: x2 + y2 + 2gx + 2fy + c = 0 is called the general equation of a circle.

The Circle
A circle is a round-shaped figure that has no corner or edges. It is a closed, two-dimensional curved shape. It is a set of all points in the plane that are a fixed point (the centre). Any interval joining a point on the circle to the centre is called a radius.

An interval joining two points on the circle is called a chord. A chord that passes through the centre is called a diameter. Since a diameter consist of two radii joined at their endpoint, every diameter has length equal to twice the radius.

A line that cuts a circle at two distinct point is called a secamt. Thus a chord is the interval cut off by a secant passing through the centre of a circle centre.

Definite Integral
In this part, the constant is removed. If a definite integration is performed, the function is evaluated between the values called limits. Upper and lower i.e. ∫3-2 (x2)dx

Example Evaluate ∫32(3x2)dx

∫3x2 = x3+c, (33 + c) - (23+c)

(27+C) - (8+C) , 27+C-8-C = 19

∫32(3x2)dx = 19

Introduction
Binomial Theorem is the method of expanding an expression that has been raised to a finite power. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

Consider the expansion of (x + y)5

(x + y)5 = (x + y)(x + y)(x + y)(x + y)(x + y)

The first term is obtained by multiplying the xs in the five brackets, there is only one way to doing this.

(x + y)n = xn + nxn-1y + n(n-1)/2! xn-2y2 + n(n-1)(n-2)/3! xn-3y3 + ...n(n-1)(n-2)...(n-r+1)/r! xn-ryr + ...yn

It can be shown that the binomial formula holds for positive, negative, integral or any rational value of n, provided there is a restriction on the values of x and y in the expansion of (x + y)n. We shall however consider only the binomial expansion formula for a positive integral n.

Introduction
Permutations and combinations are the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a Permutation while the order of selection is called Combinations.

To go further, it defines the diverse ways to arrange a certain group of data or objects. When we select the data or objects from a certain group, it is said to be Permutations, whereas the order in which they are selected is called Combination. Both concepts are very important in Further Mathematics.

Measurement of Central Tendency I
This is a measure of how the data are centrally placed. Depending on the information required, the three commonest measures of position are the arithmetic mean, median, and mode. Mean is the most widely used measure and sometimes called the arithmetical average. The mean of the number x1, x2, x3, x4 ............ xn

X = ∑x/n, where ∑x is the sum of all items. n = number of items.

When the data involves frequency; mean = ∑fx/∑

Anomalous Expansion of Water
The way substances generally react to heat is that they expand when heated as the density decreases and vice versa takes place then they are cooled. This is not the case with water. The general tendency of cold water remains unchanged until 4oC. The density of water gradually increases as you cool it. When you reach 4oC, its density reaches a maximum. When you cool water further to make ice (i.e. 0oC), it expands with a further drop in temperature; meaning the density of water decrease when you cool it form 4oC to 0oC.

The effect of this expansion of water is that the coldest water is always present on the surface which is why ice is seen on the surface of water. Since water at 4oC is the heavest, this water settles on the bottom of the water body and the lightest i.e. the coldest layer accumulates on the top layer. This is why the top of the water is always the first to freeze. Since ice and water are both bad conductors of heat, this top layer of ice insulates the rest of the water body from the cold thereby protecting all the life in the water body.

Real & Apparent Expansivity
Real expansivity of liquid is sometimes called cubic expansivity of liquid. It is defined as the increase in volume per unit degree rise in temperature. Unlike solids, liquids have no fixed length or surface area but take up the shape of the containing vessel. Therefore, in the case of liquids, we are concerned only with volume changes when they are heated.

Apparent cubic expansivity of a liquid is defined as the mass of the liquid expelled per unit divided by mass left or remaining when the temperature increases by 1oC. It is measured in K-1. Therefore,

Apparent cubic expansivity = mass of liquid expelled / (mass of liquid left x temperature rise)

Effects & Applications
- Thermometer: Thermal expansion is used in temperature measurements in thermometers.

- Bimetal strip: It consist of two thin strips of different metals such as brass and iron joined together. On heating the strip, the brass expands more than the iron. This unequal expansion causes the bending of the strip. Bimetal strips are used for various purposes; they are used to measure temperature especially in furnaces and ovens. They are used in thermostats; a bimetal thermostat is used to control the temperature of the heater coil in an electric iron.

- Riveting: To join steel plates tightly together, red hot rivets are forced through holes in the plates. The end of hot rivets is then hammered. On cooling, the rivets contract and bring the plates tightly gripped.

- Removing tight lids: To open the cap of a bottle that is very tight, it can be immersed in hot water for about a minute. This causes the metal cap to expand and become loose.

- Railway tracks: They have gaps to make allowance for expansion due to a rise in temperature, otherwise the rails would buckle.

Other effects and applications of thermal expansion are found in fixing a metallic circular strip on a wooden tyre of a cart, buckling, expansion gap, anti-scalding valve, thermostat, cooking utensils, electric fire alarm, expansion of glass, sagging of telegraph wires, railway lines, steel bridges, etc.

Note: Thermal expansion is small but it does not always have an insignificant effect.

Linear, Area & Volume Expansivity
The linear expansivity (α) of a substance is the fractional increase in its length per degree change in temperature.

α = L2 - L1/L1 ( θ2 - θ1 )

'L' denotes Length

The area expansivity (β) of a substance is the fractional increase in area per degree change in temperature.

β = A2 - A1/A1 ( θ2 - θ1 ) | β = 2α

'A' denotes Area

The volume expansivity (γ) of a substance is the fractional increase in volume per degree change in temperature.

γ = V2 - V1/V1 ( θ2 - θ1 ) | γ = 3α

'V' denotes Volume

Note: Δθ = θ2 - θ1

As the case may be, 'e' = L2 - L1 | e = Extension

As the case may be, 'e' = A2 - A1 | e = Extension

As the case may be, 'e' = V2 - V1 | e = Extension