Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere.

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If the major axis of the ellipse is along the y-axis, then the equation of the ellipse is

The locus of a point in a plane whose distance from a fixed point bears a constant ratio, less than one to its distance from a fixed line is called ellipse.

Parabolas can open rightward, open leftward, open upward and open downward.

**Focus**

The fixed point used to draw the parabola is called the focus (F). The focus is F(a, 0).

The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola. A parabola is a conic whose eccentricity is 1.

The equation Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 represents either a (non–degenerate) conic or a degenerate conic.

Let F(x_{1}, y_{1}) be the focus, lx + my + n = 0, the equation of the directrix and e the eccentricity of the conic. Let P(x, y) be any point on the conic.

The curves obtained by slicing the cone with a plane not passing through the vertex are called **conic sections** or conics.

Two circles are said to be orthogonal if the tangent at their point of intersection are at right angles.

Two circles may touch each other either internally or externally. Let C_{1}, C_{2} be the centers of the circle and r_{1}, r_{2} be their radii and P the point of contact.

The general equation of the circle is

x^{2} + y^{2} + 2gx + 2fy + c = 0

Let the line y = mx + c be a tangent to the circle x^{2} + y^{2} = a^{2} at (x1, y1)

The equation of the tangent at (x_{1}, y_{1}) to the circle x^{2} + y^{2} = a^{2} is xx_{1} + yy_{1} = a^{2}

A tangent to a circle is a straight line which intersects (touches) the circle inexactly one point.

Consider a circle with radius rand centre at the origin. Let P(x, y) be any point on the circle. Assume that OP makes an angle θwith the positive direction of x-axis. Draw the perpendicular PM to the x-axis.

A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

The equation of a pair of straight lines is in the form

**ax ^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0**, where a, b, c, f, g, h are constants.

Let the three straight lines be given by

a_{1}x + b_{1}y + c_{1} = 0

a_{2}x + b_{2}y + c_{2} = 0

a_{3}x + b_{3}y + c_{3} = 0

Let L_{1} and L_{2} be the two intersecting lines and assume that P be the point of intersection of the two straight lines which makes angle θ_{1} and θ_{2} with the positive direction of x-axis.

The length of the perpendicular from the point (x_{1}, y_{1}) to the line ax + by + c = 0 is

A straight line is the simplest geometrical curve. Every straight line is associated with an equation. To determine the equation of a straight line, two conditions are required.

If two variables, say x and y, are functions of a third variable, say θ, then the functions expressing x and y in terms of θ are called the parametric representations of x and y. The variable θ is called the parameter of the function.

The beauty and importance of the integral calculus is that it provides a systematic way for the exact calculations of many areas, volumes and other quantities.

Integration by parts method is generally used to find the integral when the integrand is a product of two different types of functions or a single logarithmic function or a single inverse trigonometric function or a function which is not integrable directly.

Sometimes, the given functions may not be in an integration form and the variable of integration (x in dx) can be suitably changed into a new variable by substitution so that the new function can be integrated. Integration by Substitution is also called **u-substitution**.

This rule applies to functions of ax+b. It does not apply in other cases.

If ∫ f(x) dx = F(x) + C, then

**∫ f(ax + b) dx = 1/a × F(ax + b) + C**

Integration is the inverse process of differentiation. A function F(x) is called an anti derivative or integral of a function f(x) on an interval I if

F′(x) = f(x) for every value of x in I

If u = f(x) and y = F(u), then y = F(f(x)) is the composition of f and F. u is called the **intermediate argument**.

If u and v are differentiable function and if v(x) ≠ 0, then

Let u and v be differentiable functions of x. Then the product function

y = u(x) v(x) is differentiable.

Consider a function y = f(x) of a variable x. Suppose x changes from an initial value x_{0} to a final value x_{1}. Then, the increment in x is defined to be the amount of change in x. It is denoted by ∆x.

The derivative of a given function y = f(x) is defined as the limit of the ratio of the increment ∆y of the function to the corresponding increment ∆x of the independent variable, when the latter tends to zero.

Let f be a function defined on an interval I = [a, b]. A continuous function on I is a function whose graph y = f(x) can be described by the motion of a particle travelling along it from the point (a, f(a)) to the point (b, f(b)) without moving off the curve.

The notion of limit is related to the intuitive idea of nearness or closeness. Degree of such closeness cannot be described in terms of basic algebraic operations of addition and multiplication and their inverse operations subtraction and division respectively.

If f(x) = f(−x) for all x in the domain then the function is called an **even function**.

If f(x) = − f(−x) for all x in the domain then the function is called an **odd function**.

The sign function or signum function extracts the sign of a real number.

The function whose value at any real number x is the greatest integer less than or equal to x is called the greatest integer function (floor function). It is denoted by [x].

If f : R → R defined by f(x) = |x| then the function is called absolute value function of x.

For any number a > 0, a ≠ 1, the function f : R → R defined by **f(x) = a ^{x}** is called an exponential function.

If a function f : R → R is defined in the form **f(x) = ax + b** then the function is called a linear function. Here, a and b are constants.

If the range of a function is a singleton set then the function is called a constant function.

To define the inverse of a function f i.e. f^{−1} (f inverse), the function f must be one-to-one and onto.

Let A, B and C be any three sets and let f : A → B and g : B → C be any two functions. The domain of g is the co-domain of f.

A function f from a set A to the same set A is said to be an identity function if f(x) = x for all x ∈ A i.e. f : A → A is defined by f(x) = x for all x ∈ A. Identity function is denoted by I_{A} or I. Therefore, I(x) = x always.

A function is said to be one-to-one if each element of the range is associated with exactly one element of the domain. Two different elements in the domain (A) have different images in the co-domain (B).

If the range of a function is equal to the co-domain then the function is called an **onto function**. Otherwise it is called an **into function**.

A function is a special type of relation. In a function, no two ordered pairs can have the same first element and a different second element. That is, for a function, corresponding to each first element of the ordered pairs, there must be a different second element.

A quantity which retains the same value throughout a mathematical process is called a **constant**. A **variable** is a quantity which can have different values in a particular mathematical process.

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