Solutions / Properties of triangles for IIT-JEE MAINS /ADVANCED

Video lectures  of KAMAL SIR  for  Properties of triangles

 

 

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In a triangle ABC the angles are denoted by capital letters A, B and C and the length of the sides opposite to these angles are denoted by small letters a, b and c. Semi perimeter of the triangle is
given by s where  2s = a + b+c  and its area is denoted by Δ

SINE  RULE

Sine rule is an  important tool that relates   side lengths with angles of triangles  and circum radius of triangle

PROPERTIES OF TRIANGLE 1

NAPIER’S ANALOGY (TANGENT RULE)

PROPERTIES OF TRIANGLE

COSINE RULE
In a triangle ABC

properties of triangle -- cosine rule
properties of triangle — cosine rule

 

PROJECTION RULE

(i) a = b cos C + c cos B (ii) b = c cos A + a cos C (iii) c = a cos B + b cos C

HALF ANGLE FORMULAE

PROPERTIES OF TRIANGLE

 

.m-n THEOREM

PROPERTIES OF TRIANGLES

PROPERTIES OF TRIANGLES

 

CENTROID AND MEDIANS OF A TRIANGLE
The line joining any vertex of a triangle to the mid point of the opposite side of the triangle is called the median of the triangle. The three medians of a triangle are concurrent and the point of concurrency of the medians of any triangle is called the centroid of the triangle. The centroid divides the median
in the ratio 2 : 1.

 

Circum circle

PROPERTIES OF TRIANLES

The circle which passes through the angular points of a ABC, is called its circumcircle. The centre of this circle i.e., the point of concurrency of the perpendicular bisectors of the sides of the ABC, is called the circumcenter.

Radius of the circumcircle is given by the following formulae

PROPERTIES OF TRIANGLE
ORTHOCENTER AND PEDAL TRIANGLE OF A TRIANGLE.
In a triangle the altitudes drawn from the three vertices to the opposite sides are concurrent and the point of cuncurrency of the altitudes of the triangle is called the orthocenter of the triangle. The triangle formed by joining the feet of these perpendiculars is called the pedal triangle i.e.
DEF is the pedal triangle of ABC.

PROPERTIES OF TRIANGLES

 

BISECTORS OF THE ANGLES
If AD bisects the angle A and divide the base into portions x and y, we have, by Geometry,
properties of triangle

 

INCIRCLE
the circle which can be inscribed within the triangle so as to touch each of the sides of the triangle is called its incircle. The centre of this circle i.e., the point of concurrency of angle bisectors of the triangle is called the incentre of the ABC

properties of triangle

The distances Between the special points
properties of triangle

ESCRIBED CIRCLES
The circle which touches the side BC and the two sides AB and AC produced is called the escribed circle opposite the angle A. Its centre and radius will be denoted by I1 and r1 respectively.

properties of triangle

Excentral triangle
The triangle formed by joining the three excentres I1, I2 and I3 of
DABC is called the excentral or excentric triangle. Not that

properties of triangles

Inscribed & Circumscribed Polygons
(Important Formulae)

sot 46

 

SOLUTION OF TRIANGLES
When any three of the six elements (except all the three angles) of a triangle are given, the triangle is known completely. This process is called the solution of triangles.

sot 47

sot 48

 

Complex numbers for IIT-JEE MAINS /ADVANCED

Complex numbers basic concepts-Defining

A number in the form of a + ib, where a, b are real numbers and is called a complex
Number. A Complex Number can also be defined as an ordered pair of real numbers a and b any
may be written as (a, b), where the first number denotes the real part and the second number
denotes the imaginary part. If z = a + ib, then the real part of z is denoted by Re (z) and the
imaginary part by Im(z). A complex number is said to be purely real if Im(z)= 0, and is said to
be purely imaginary if Re(z) = 0. The complex number 0 = 0 + i0 is both purely real and imaginary.
Two complex numbers are said to be equal if and only if their real parts and imaginary parts are
separately equal i.e. a + ib = c + id implies a = c and b = d. However, there is no order relation
between complex numbers and the expressions of the type a + ib < (or > ) c + id are meaningless

Geometrical representation of complex numbers

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes.

complex numbers 1

The plane of OX and OY is called the Argand diagram or the complex plane. Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.

 

Modulus and Argument of a Complex Number

We define modulus of the complex number z = x + iy as

complex numbers 2

complex numbers 3
We can define the argument of a complex Number also as any value of the which satisfies the
complex numbers 4
is the angle which OP makes with the positive x-axis.

Trigonometric ( or polar ) from a Complex Number

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complex numbers 6

Remark : Method of finding the principal value of the argument of a complex number z = x + iy.

complex numbers 7

Unimodular Complex Number

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Algebraic Operations with Complex Number

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Geometrical Meaning of Product and Division

Let z1 = x1 + i y1 and z2 = x2 + i y2 be two complex numbers represented by the points P1(x1, y1) and P2(x2, y2) respectively. By definition z1 + z2 should be represented by the point (x1 + x2 , y1 + y2 ). This point is the vertex which completes the parallelogram with the line segments joining the origin with OP1 and OP2 as the adjacent sides.
| z1 + z2 | = OP.

complex numbers 10
lso by definition z1 – z2 should be represents by the point (x1 – x2, y1 – y2). This point is the vertex
which completes the parallelogram with the line segments joining the origin with OP1 and
(where the point represents ; the point can be obtained by producing the directed
line P2O by length | z2 | ) as the adjacent sides.
| z1 – z2 | = OQ = P2 P1.
Remarks
(a) In any triangle, sum of any two sides is greater than the third side and difference of any two

complex numbers 11

Construction for the point representing the product z1 z2

complex numbers 12

complex numbers 13

.

Construction for the point representing the quotient z1/z2

Draw the triangle OQ1P directly similar to the triangle OQ2L
The P represents the quotient z1/z2 .

complex numbers 14

complex numbers 15

 

Square Root of a Complex Number

How we get the square root of complex number

complex-numbers-16

complex numbers 17

Conjugate of a Complex Number

The conjugate of the complex number z = a + ib is defined to be a – ib and i denoted by . In other words is the mirror image of z in the real axis.

complex numbers 18

Properties of Conjugate, Modulus, Argument

complex numbers 20

complex numbers 21

complex numbers 22
Modulus of a Complex Number

Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| .

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complex numbers 24

Properties of Arguments

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Complex Numbers Represented By Vectors
It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors . The addition or the subtraction of two complex numbers is also the same as the addition or the substraction of two vectors. This fact is fundamental in theory and very useful in practice.
It should be noticed that if a number z is represented by points P and OP by a vector then
| z | is the length OP and arg(z) is the angle which the directed line OP makes with directed OX.
Please note that if z = x + iy and P is the point (x, y), a one–to–one correspondence exists between the number z and any of the following : (i) the point P; (ii) the displacement ; (iii) the vector
(or directed length ) .
Any one of these three things may therefore be said to represent z, or to be represented by z.

 

De-Moiver’s Theorem

complex numbers 27

Application of De Moiver’s Theorem

This is a fundamental theorem and has various applications. Here we will discuss few of these
which are important from the examination point of view

The nth Roots of Unity

complex numbers 28
Cube Roots of Unity

complex numbers 30

Logarithm of Complex Number

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CONCEPT OF ROTATION

complex numbers 32

complex numbers 33

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Geometrical Application

Section Formula

complex numbers 35

Equation of a straight line

Equation of straight line with the help of coordinate geometry

complex numbers 36
Equation of straight line with the help of rotation formula
complex numbers 38
General equation of the line

complex numbers 39

complex numbers 40

complex numbers 42

 

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Equation of Perpendicular Bisector

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arg (z – z0) =p  represents a line passing through z0 with slope (making angle p with the positive direction of x-axis).

Equation of a circle

Consider a fixed complex number z0 and let z be any complex number which moves in such a way
that it’s distance from z0 is always equals to ‘r’. This implies z would lie on a circle whose centre is
z0 and radius r. And its equation would | z – z0 | = r

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It represents the general equation of a circle in the complex plane.

complex numbers 46

complex numbers 47

complex numbers 48

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Note :
(i) If we take ‘C’ to be mid–point of A2A1, it can be easily proved that CA.CB = (CA1)2 i.e. | z1 – z0 | | z2 – z0 | = r2, where the point C is denoted z0 and r is the radius of the circle.

(ii) If | z1 – z0 | = | z2 – z0 | hence P(z) would lie on the right bisector of the line A(z1)
and B(z2). Note that in this case z1 and z2 are the mirror images of each other with respect
to the right bisector.

Equation of tangent a given circle at a given point

complex numbers 50

Some important results

complex numbers 51

ENJOY LEARNING MATHS WITH VIDEO LECTURES OF KAMAL SIR 

kamal sir > Well - Known IIT JEE Maths mentor

kamal sir > Well – Known IIT JEE Maths mentor