PARABOLA
1. CONIC SECTIONS:
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
a) The fixed point is called the focus.
b) The fixed straight line is called the directrix .
c) The constant ratio is called the eccentrity denoted by e.
d) The line passing through the focus and perpendicular to the directrix is called the .
e) A point of intersection of a conic with its axis is called a vertex .
2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY:
The general equation of a conic with focus (p,q) and directs lx + my+ n= 0 is:
(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)²≡ ax²+ 2hxy + by²+ 2gx + 2fy + c=0
3. DISTINGUISHING BETWEEN THE CONIC:
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.
Case (i) : when the focus lies on the directrix:
In this case D ≡ ABC + 2 fgh - af²- bg²- ch²= 0 and the general equation of a conic represents a pair of straight lines and if:
e> 1 the lines will be real and distinct intersecting at S.
e= 1 the lines will coincident.
e< 1 the lines will be imaginary.
Case (ii): when the focus does not lie on the directrix :
The chronic represents :
A parabola : e= 1; D≠ 0; h½= ab
An ellipse : 0< e < 1; D≠ 0; h² < ab
A hyperbola: D≠ 0; e> 1; h² > ab
A rectangular hyperbola : e> 1; D ≠ 0 ; h² > ab ; a+ b = 0
PARABOLA
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed (focus) is equal to its perpendicular distance from a fixed line (directrix).
Standard equation of the parabola is y²= 4ax. For this parabola :
i) vertex is (0,0)
ii) Focus is (a,0)
iii) Axis is y= 0
iv) Directrix is x + a = 0
a) Focal distance:
The distance of a point in the parabola from the focus is called the focal distance of the point.
b) Focal chord:
A chord of the parabola, which passes through the focus is called a focal chord.
c) Double ordinate:
A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.
d) Latus rectum:
A double ordinate passing through the focus or a focal chord perpendicular to axis of parabola is called the latus rectum . For y²= 4ax.
• Length of the latus rectum= 4a.
• Length of the semi latus rectum= 2a.
• Ends of the latus rectum are L(a, 2a) and L'(a, -2a)
Note that:
i) Perpendicular distance from focus on directrix = half the latus rectum.
ii) Vertex is middle point of the focus and the point of intersection of the directrix and axis.
iii) Two parabolas are said to be equal if they have the same latus rectum.
5. PARAMETRIC REPRESENTATION:
The simple and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² and y= 2at together represents the parabola y²= 4ax, t being the parameter.
6. TYPE OF PARABOLA :
Four standard forms of the parabola are y²= 4ax; y²= - 4ax; x²= 4ay; x²= - 4ay
DEFINITION:
A Set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is call ped Matrix.
ORDER OF A MATRIX
A matrix having m rows and n columns is called a of order mxn.
a₁₁ a₁₂ a₁₃ ....a₁ⱼ ....a₁ₙ
a₂₁ a₂₂ a₂₃.....a₂ⱼ.....a₂ₙ
| | | | |
aᵢ₁ aᵢ₂ aᵢ₃......aᵢⱼ......aᵢₙ
aₘ₁ aₘ₂ aₘ₃ ....aₘⱼ.....aₘₙ]ₘₓₙ
Or A= [aᵢⱼ]ₘₓₙ, 1≤ i≤ m, 1 ≤ j≤ n, i,j ∈ N
TYPES OF MATRIX
1. Rectangular matrix : If the number of rows in the columns of a matrix are not equal, then the matrix is called rectangular Matrix, i.e., m≠ n
2. Row Matrix : A matrix containing only one row is called a row matrix. i.e., m= 1
3. Column Matrix: A matrix containing only one column is called a column matrix i.e., n= 1
4. Null or Zero Matrix : If every element of a matrix is zero, then it is called a zero or null matrix.
5. Square matrix: A matrix in which the number of rows is equal to the number of columns is called square Matrix p.
6. Diagonal Matrix: A square Matrix in which every non diagonal element is zero is called a diagonal Matrix.
7. Scalar matrix : A square Matrix in which all the elements in the diagonals are equal and rest of the elements are zero is called a scalar matrix.
8. Identity or unit matrix: A square in which each diagonal elements is unity and all other elements are zeros is called a unit matrix. It is denoted by I.
9. Upper triangular matrix : A square Matrix in which all the elements below the diagonal are zeros is called an upper triangular matrix.
10. Lower triangular matrix: A square Matrix in which all the elements above the diagonal are zeros is called a lower triangular matrix.
11. Equal matrix : Two matrices are said to be equal if they are of the same order and each element of one matrix is equal to the corresponding element of other matrix.
OPERATION ON MATRICES
• Matrix addition : Let A= [aᵢⱼ] and B=[bᵢⱼ] be two mxn matrices. Then the sum A+ B of A and B is defined as A+ B= [cᵢⱼ]ₘₓₙ , where cᵢⱼ = aᵢⱼ + bᵢⱼ for each i and j.
• Substraction of a matrix : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₘₓₙ be two matrices of same order, then the difference A - B of A and B is defined as A - B = A +(- B)= [aᵢⱼ - bᵢⱼ]ₘₓₙ for each i and j.
• Negative of Matrix : Let A= [aᵢⱼ]ₘₓₙ be a matrix, then the negative of the matrix A is defined as the matrix [-aᵢⱼ]ₘₓₙ and is denoted by - A. Thus negative of a matrix is obtained by making all the elements of the given matrix negative .
• Scalar multiplication of a matrix: Let [aᵢⱼ]ₘₓₙ be a matrix and k is a scalar, then the Matrix obtained by multiplying the individual element of the matrix A by k is called the scalar multiplication of A by k and is denoted by kA or Ak.
• Multiplication of two matrices : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₙₓₚ be two matrices, then the product AB= C = [cᵢₖ]ₘₓₚ where cᵢₖ = aᵢ₁ b₁ₖ + aᵢ₂ b₂ₖ + ....+ aᵢₙ bₙₖ.
Properties of matrix Addition and Multiplication
• Commutative law
If A and B are two matrices, then
A+ B= B+ A and AB ≠ BA
•Associative law
If A, B and C are three matrices, then
(A+ B)+ C = A+(B + C) and (AB)C = A(BC(
• Distributive law
A(B+ C)= AB+ AC and (A+ B)C = AC+ BC
• Existence of Identity
The null matrix is the identity element for Matrix edition i.e., A+ O = A= O+ A
If A is a square Matrix, then IA = A = AI, where I is identity matrix of same order.
• Existence of inverse :
For every matrix A= [aᵢⱼ]ₘₓₙ there exists a matrix [-aᵢⱼ] ₘₓₙ denoted by - A, such that A+ (-A)= O= (-A)+ A
• If A is mxn matrix and O is a null Matrix, then
i) Aₘₓₙ Oₙₓₚ = Oₘₓₚ
ii) Oₚₓₘ Aₘₓₙ = Oₚₓₙ
TRANSPOSE OF A MATRIX
• Let A= [aᵢⱼ]ₘₓₙ matrix. Then the transpose of A, denoted by Aᵀ or A', is an nxm matrix such that Aᵀ= [aᵢⱼ]ₘₓₙ and i= 1,2,....m and j= 1,2,....,n.
Thus Aᵀ is obtained from A by changing its rows in to columns and its columns into rows.
Properties of Transpose
• Let A and B be two matrices, then
i) (Aᵀ)ᵀ= A
ii) (A+ B)ᵀ= Aᵀ + Bᵀ,where A and B being of the same order.
iii) (kAᵀ)= kAᵀ, k be any scalar (real or Complex)
iv) (AB)ᵀ = BᵀAᵀ , A and B being conformable for product AB . (Reversal law)
v) (ABC)ᵀ= CᵀBᵀAᵀ.
ELEMENTARY OPERATION
There are six operations (transformations) on a matrix , three of which are due to rows and three due to columns, which are known as alimentary operations or transformations .
i) The multiplication of the elements of any row or column by a non zero number. Rᵢ --> Rᵢ
The corresponding column operation is denoted by Cᵢ --> kCᵢ where k≠ 0
For example, applying R₂ --> (1/4) R₂ to A
A= a₁₁ a₁₂ we get a₁₁ a₁₂
a₂₁ a₂₂ a₂₁/4 a₂₂/4
ii) The interchange of any two rows or two columns i.e., Rᵢ <---> Rⱼ and Cᵢ <--> Cⱼ.
For example, applying R₁ <--> R₃ to A
a₁₁ a₂₁ a₁₃ a₃₁ a₃₂ a₃₃
A=a₂₁ a₂₂ a₂₃ we get a₂₁ a₂₂ a₂₃
a₃₁ a₃₂ a₃₃ a₁₁ a₁₂ a₁₃
iii) The addition to the elements of any two rows or column, the corresponding elements of any other row or column multiplied by any non-zero number
i.e., Rᵢ---> Rᵢ + kRⱼ or Cᵢ---> Cᵢ + kCⱼ.
For example applying C₁--> R₁ - kR₂ to A
A= a₁₁ a₁₂ we get a₁₁ - ka₂₁ a₁₂ - ka₂₂
a₂₁ a₂₂ a₂₁ a₂₂
SOME SPECIAL MATRICES
• symmetric matrix : A square Matrix A= [aᵢⱼ] is called a symmetric matrix if aᵢⱼ = aⱼᵢ for all i, j.
• Skew-symmetric matrix: A square matrix A= [aᵢⱼ] is a Skew-symmetric matrix if aᵢⱼ = - aⱼᵢ for all i, j.
• Orthogonal matrix : A square matrix A is called an orthogonal matrix if AAᵀ =AᵀA = I.
• Equivalent matrices: Two matrices A and B are equivalent if one can be obtained from the other by a sequence of elementary row transformations .
INVERTIBLE MATRICES
if A is a square matrix of order n and if there exists another square Matrix B of the same order n, such that AB = BA= I, then B is called the inverse matrix of A and it is denoted by A⁻¹. In that case A is said to be invertible.
Uniqueness of inverse
Inverse of a square matrix, if it exists , is unique .
Let A=[aᵢⱼ] be a square matrix of order n. If possible, let B and C be two inverses of A.
To show: B= C
Since B is the inverse of A
=> AB= BA = I
Since C is also the inverse of A
=> AC= CA= I
Thus B= BI = B(AC)= (BA)C = IC = C
If A and B are invertible matrices of the same order, then (AB)⁻¹= B⁻¹A⁻¹.
Inverse of a Matrix by Elementary Operation
To find A⁻¹.using elementary row operations , write A= IA and apply a sequence of row operations on A= IA till we get, I= BA. The matrix B will be the inverse of A. Similarly, if we wish to to find A⁻¹ using column operations , then write A= AI and apply a sequence of colum operations on A= AI till we get, I= AB
DETERMINANTS
DEFINITION: Corresponding to each square Matrix A= (aᵢⱼ)ₘₓₙ there is an associated expression , called the determinant of A, denoted by det A or |A|, written as
|a₁₁ a₁₂ a₁₃ ....a₁ₙ
a₂₁ a₂₂ a₂₃.....a₂ₙ
.... .... ... ... ...
aₙ₁ aₙ₂ aₙ₃ ....aₙₙ|ₘₓₙ
• Value of determinant of order 1
Let A= [a] be Matrix of order 1, then determinant A is defined to be equal to a.
• Value of determinant of order 2
|a₁₁ a₁₂
a₂₁ a₂₂| = (a₁₁ a₂₂ - a₂₁ a₁₂)
• Value of a determinant of order 3 or more
The value of a determinant is the sum of the product of a row (or a column) with their corresponding factors.
So, det A= ⁿᵢ₌₁∑ aᵢⱼ Cᵢⱼ
• Properties of determinant
i) det Iₙ= 1, where Iₙ is unit/identity where unit/identity matrix of order n.
ii) det 0ₙ= 0, 0 is null Matrix (square matrix of any order).
iii) A= (aᵢⱼ)ₙₓₙ then |A|= |A'| (Reflection Property).
iv) det (AB )= det A . det B, where A & B are matrices of same order.
v) det(kA)= kⁿ det A, if A is of order n x n.
vi) det (Aⁿ)= (det A)any if n ∈ I⁺
vii) |A|= 0 iff
a) Any two rows or columns are identical.
b) Any two rows or columns are in proportion.
c) Each elements of any row/ column is zero.
viii) if each element of a row/column of a determinant is multiplied by k then value of new determinant is k times the original Determinants.
ix) Determinant of a diagonal matrix is the product of its diagonal elements.
x) If two row/columns of a determinant are interchanged, then the determinant retains its absolute value but its sign is changed.
xi) if each element of a row/columns of a determinant is expressed as a sum of two or more terms, then determinant can be expressed as the sum of two or more determinants.
xii) if any row/column of a determinant, a multiple of another row/column is added, then the value of determinant does not change.
xiii) The sum of product of the elements of any row/column of a determinant with cofactors of the corresponding elements of any other row/ column is zero.
APPLICATION OF DETERMINANTS
• Area the triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by
(1/2) | x₁ y₁ 1
x₂ y₂ 1
x₃ y₃ 1
• If area is zero then there points are collinear.
MINORS AND COFACTORS
• Minor of aᵢⱼ in |A|: The minor of an element aᵢⱼ in |A| is defined as the value of the determinant obtained by deleting the iᵗʰ row and jᵗʰ column of |A|, and it is denoted by Mᵢⱼ .
• Cofactor of aᵢⱼ in |A|: The cofactor Cᵢⱼ of an element aᵢⱼ is defined as Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ.
ADJOINT OF A MATRIX
• Adjoint of a matrix A= (aᵢⱼ)ₙₓₙ is defined as adj A= [Cᵢⱼ]ₙₓₙ, where
Cᵢⱼ represents cofactor of aᵢⱼ in |A|.
Properties of Adjoint of a matrix
• For every square matrix
A(adj A)= (adj A)= |A|. I.
• If |A|=0 matrix A is called singular else non singular.
• if A and B are non singular matrices of the same order, then AB and BA are also non singular matrices of same order.
• (adj AB)= (adj B).(adj A)
• (adj A)' = adj A'.
• let A be n x n matrix, then
i) |adj A|= |A|ⁿ⁻¹
ii) adj (adj A)= |A|ⁿ⁻²A
iii) |adj (adj A|= |A|(ⁿ⁻¹)^2
INVERSE OF A MATRIX
A non zero matrix A of order n is said to be invertible if there exists a square matrix B of order n such AB = BA= I. We say A⁻¹= B.
• A matrix A is invertible if |A|≠ 0.
• A⁻¹= 1/|A|. (adj A).
• If AB= AC => B = C if |A|≠ 0.
• (AB)⁻¹= B⁻¹A⁻⅓.
• (Aᵀ)⁻¹= (A⁻¹)ᵀ.
• If A is an invertible symmetric matrix then A⁻¹ is also symmetric.
SOLUTION OF SYSTEM OF EQUATIONS
• let AX = B be the given system of equations:
i) If |A|≠ 0, the system is consistent and has one unique solution.
ii) If |A|= 0 and (adj A)B ≠ 0, then the system is inconsistent so it has no solution.
iii) If |A|= 0 and (adj A)B=0, then the system is consistent but has infinitely many solutions.
TANGENT & NORMAL
Things to remember:
CIRCLE
Circle is defined as a local self found in the Cineplex as the distance from a fixed point in the plane is constant the fixed point is called the centre of the circle on the constant distance is called the radius of the circle standard equation of a circle Central forms represent the circle with centre in the radius of the centre is at 007 circle become circle passing through the origin circle touching circle passing through the origin and Central line on the exactis and circle passing to the original Central line on the y-axis general equation the circle is equation of a circle centre in the radius and for a real circle and for the Point Circle and forum in imaginary circle conditions of a circle it is a second degree Celsius equation there is no time containing it contest we are literally constant for the circle refers to write as Centre is under radius equation of a circle when coordinates are pain the point of the time which are position of a point a circle the point lies outside on or inside a circle according circleall the pragmatic position of we write the points are the parameter equation normal atomic count On the circle is a straight line which is perpendicular to the tangent with the curve at the point length of the tangent from to the circle pair of tangent drawn from a point to a given circle is given by circle is similarly for the circle director circle the locus of the point of intersection at 2 perpendicular timing to a given circle is called a director circle is a director circle for circle the condition that to circle should interface area and sufficient condition for the surprise to interested to distinct points is where with the centres with the radii of the two circles external and internal contact of the circles to circle with centre and ready respectively touch each other internallength of an external or direct common time zone or transversal common timing to length of a external on direct common tangent and internal transfer common tangent to the two circle is given by were distance between the centre of the circles are there of the two circles the length of the internal common tangent is always less than length of the external or direct common tangent the direct common timing to circle the transversion common Tangled also meet on the line centre and divided internal on the ratio of the riding chord of the contact of two tangents drawn from to the circle similarly for the circle equation the chord with given midpoint is for the circle equation of the chords with midpoint similarly for the general circle common code the to circle the core joining the points of intersection of two given circle is called their common cause if denote the centre of the given circle then the order radius of the circle length of the perpendicular from the common cause of the circle to circle are safety after 1 time to be circles at either point of section are at right anglecondition for two circles the orthogonal is ID collecting the focus in such a way so that length of tangent drawn from it to the two circles are equal is straight line known as the radical accessed of the given to circles a part 2 circles the radical exercise given by any circle passing through the point of intersection of circle where is constant any circle passing through a point of intersection
PROBABILITY
Random experiment and operation which can result in any one app its weight outcomes and outcom cannot be predicted its certainty sample space on a sample point the set of all possible outcomes around experimental sample space and is denoted by a brief possible outcomes April point trial and experimental repeat under the similar code does not give the same result it's time a subset sample space setup some possible outcomes a random experiment simple in bank contains only one sample points in banks contains no sample points missed compound more than one sample point complement of an agent contents all the sample point other than those content in event the union and the consist of all points are both the intersection and concerts of all points that are in both and and have a no points in common we write the complement for consist of all the points of not in the sequel like less none of them is expected to occur in preference in othersmutually exclusive events occurrence of one of them precludes are current affairs exhaustiblements performance of the experiment result in occurrence of at least one of them binomial distribution out of independence trials the probability of getting success at least success at most success where is the probability of success and probability classical definition basicap probability for an event is complement in sample space for event for exhaustive banks in a sample space for mutually exclusive a balance in a sample space for mutual exclusive and exhaustive events for an event in sample space the odds in favour where is the complement of the event also against conditional probability the conditional probability of given is independent event events are independent if the reference does not affect each other and also they must be there wise independent total probability in the best theorem in a sample space to be the set of mutual explosion exhaustive set of sample space the total probability of a band is given by the an event of which has already occurred than condition probability of occurrence of any one of the event out of 1 to
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