Simple math, Complex life

Definition :          The ellipse is the intersection of double napped cone with an oblique plane. An ellipse is the locus of a point in a plane which moves so that its distance from a fixed point bears a constant ratio e (0 < e < 1) to its distance from a fixed line. The fixed point is called the focus S and the fixed line is callled the directrix d . If S is a fixed point is called focus and directrix d is a fixed line not containing the focus then by definition PS/PM = e and PS = e PM  , where PM is the perpendicular on the directrix and e is the real number with 0 < e < 1 called eccentricity of the ellipse. Standard Equation of Ellipse :       Let's derive the standard equation of the ellipse  Let S be the focus , d be the directrix and e be the eccentricity of an ellipse. Let  SZ perpendicular to directrix d . A ans A ' divide the segment SZ internaly as well as externally in the ratio e : 1 . Let AA ' = 2a , midpoint O of segment AA ' be the origine .

        The hyperbola is the intersection of double napped cone with plane parallel to the axis.  The hyperbola is the locus of a point in a plane which moves so that its distance from a fixed point bears a constant ratio e (e > 1) to its distance from a fixed line . The fixed point is called the focus S and the fixed line is called the directrix d .                        If S is the focus and d is the directrix not containing the focus and P is the moving point, then PS/PM = e , where PM is the perpendicular on thr directrix . e > 1 called eccentricity of the hyperbola   Standard Eqaution of The Hyperbola :       Let S be the focus, d be the directrix and e be the eccentricity of a hyperbola . Draw SZ perpendicular to directrix . Let A and A ' divide the segment SZ internally and externally in the ratio e:1. By definition of hyperbola A and A ' lie on hyperbola. Let AA ' = 2a , midpoint O of segment AA' be the origine . Then O ≡ (0, 0) , A≡ (a, 0) and A ' ≡

Definition :   A parabolla is the locus of the point in plane equidistant from a fixed point and a fixed line in that plane . The fixed point is called the focus and the fixed straight line is called the directrix. Standard Equation Of The Parabola : Equation of the parabola in the standard form y 2  = 4ax. Let S be the focus and d be the directrix of the parabola . Let SZ be perpendicular to the directrix . Bisect SZ at the point O. By the definition of parabola the midpoint O is on the parabola. Take O as the origine , line OS as the X- axis and and the line through O perpendicular to OS as the Y-axis .        Let SZ = 2a , a > 0 . Then the coordinates of the focus S are (a, 0) and the coordinates of Z are (-a, 0). The equation of the directrix d is  x = -a , i.e. x + a = 0 Let P(x, y) be any point on the parabolla . Draw segment PM perpandicular to the directrix d . ∴ M = (-a, y)  By using distance formula we have By focus - directrix property of the parabolla SP = PM Squaring bo

 The greek mathematician Archimedes and Apollonius studied the curve named conic sections . These curve are intersection of a plane with right circular cone. Conic section have a wide range of applications such as planetary motions, in desigens of telescope and antennas, eflection in flash , automobile headlights, construction of bridge , navigation, projectiles etc. Definition of conic section:     A conic section or conic can be defined as the locus of the P in a plane such that the ratio of the distance of P from a fixed point to its distance from a fixed line is constant  e known as  eccentricity .     The fixed point is callled the focus of the conic section, denoted by S . The fixed straight line is called the directrix of conic section, denoted by  d. If S is the focus , P is any point on the conic section and segment PM is the length of perpandicular from P on the derectrix, then by definition SP/PM = constant.              The nature of the conic section depends upon the value

    Proportion : Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other. Proportions are denoted by the symbol  ‘::’ or ‘=’. The proportion can be classified into the following categories, such as: Direct Proportion Inverse Proportion Continued Proportion Direct Proportion The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b. Inverse Proportion The inverse proportion describes the relationship between two quantities in which an increase i

Rational Numbers:       The rational number is expressed as frational form i.e. m/n are callled rational numbers. Here , m and n are integers but n is not equal to zero . eg., 5/2 is a rational number it means that 5 is divided by 2 . Comparison of Rational numbers :         For any pair of number line the number to the lest is smaller that the other. Also , if the numeraror and the denominator of a rational number is multiplied by any non zero number then the value of rational number does not change. It remains the same . That is a/b = ka/kb , (k ≠ 0). eg ., Compare the number 5/4 and 2/3 . write using the proper symbol of <, > , =. solution :     5/4  = 5 х 3 / 4 х 3  = 15/12     2/3  = 2 х 4 /3 х  4 =  8/12      15/12  >  8/12    ∴  5/4  > 2/3   Decimal Representation of  Rational Numbers :      If we use decimal fractions which dividing the numerator of a rational numbe by its denominator , we get the decimal representation of a rational numbe . For example 5/2 = 2.5 .

 The concept of matrices was developed by a Mathematician Arthur Cayley . Matrices are useful in expressing numerical information in compact form. They are effectively used in expressing different operators . Definition :      A rectangular arrangememnt of mn numbers in m rows and n columns, emclosed in [ ] or ( ) is called a matrix of order m by n .  A matrix by itself does not have a value or any special meaning . Order of the matrix is denoted by m m🗙n , read as m by n . Each member of the matrix is called an element of the matrix . Matrices are generaly denoted by A, B, C, ... and their elements are denoted by a ij , b ij   ,  c ij , ...etc. Matrix is generaly written as : Important Formulas for  Matrices  If A and B are square matrices of order n, and I n  is a corresponding unit matrix , then (a)  A(adj.A) = | A | I n  = (adj A) A (b)  | adj A | = | A |n -1  (Thus A (adj A) is always a scalar matrix) (c)  adj (adj.A) = | A | n-2  A (d)  adj (AB) = (adj B) (adj A) (e)  adj (A m