Video lectures of KAMAL SIR forProperties of triangles
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 is an important tool that relates side lengths with angles of triangles and circum radius of triangle
NAPIER’S ANALOGY (TANGENT RULE)
In a triangle ABC
(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
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.
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
BISECTORS OF THE ANGLES
If AD bisects the angle A and divide the base into portions x and y, we have, by Geometry,
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
The distances Between the special points
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.
The triangle formed by joining the three excentres I1, I2 and I3 of
DABC is called the excentral or excentric triangle. Not that
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.
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
Trigonometric ( or polar ) from a Complex Number
Remark : Method of finding the principal value of the argument of a complex number z = x + iy.
Unimodular Complex Number
Algebraic Operations with Complex Number
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.
Construction for the point representing the product z1 z2
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 .
Square Root of a Complex Number
How we get the square root of complex number
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.
Properties of Conjugate, Modulus, Argument
Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| .
Properties of Arguments
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
Logarithm of Complex Number
CONCEPT OF ROTATION
Equation of a straight line
Equation of straight line with the help of coordinate geometry
Equation of Perpendicular Bisector
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
(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
The quadratic equation has real and equal roots if and only if D = 0
The quadratic equation has real and distinct roots if and only if D > 0
The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 . If p + iq (p and q being real) is a root of the quadratic equation where i = , then p – iq is also a root of the quadratic equation.
CONCEPT OF IDENTITY
If the quadratic equation is satisfied by more than two distinct numbers (real or complex), then it becomes an identity i.e. a = b = c = 0. For example
is satisfied by three value of x which are a, b and c. Hence this is an identity in x.
HOW TO SOlVE RATIONAL INEQUALITIES
In order to solve inequalities of the form
we use the following method:
If x1 and x2 (x1 < x2) are two consecutive distinct roots of a polynomial equation, then within this
interval the polynomial itself takes on values having the same sign. Now find all the roots of the polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write
Other way of understanding the same concept
Write the inequality in the correct form. One side must be zero and the other must be in product form
Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and change them in product of linear forms
Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to and mention them on line in increasing form.
Perform the sign analysis. and make wave that give the sign of product
What is needed acording to that write final answers
Problems related to rational inequalities with video solutions
CONDITIONS OF COMMON ROOTS FOR TWO QUADRATIC EQUATIONS
For two quadratic equations when one root is common / When both roots are common are given
HOW TO FIND RANGE OF A RATIONAL EXPRESSION IN X, WHERE X IS REAL
Put the given rational expression equal to y and form the quadratic equation in x.
Find the discriminant D of the quadratic equation obtained in step 1.
Since x is real, therefore, put D GREATER THAN OR EQUAL TO 0. We get an inequation in y.
Solve the above inequation for y. The range of y so obtained determines the range attained by the given rational expression
QUADRATIC EXPRESSION-LOCATION OF ROOTS
The expression ax2 + bx + c is said to be a real quadratic expression in x where a, b, c are real and
a 0. Let f(x) = ax2 + bx + c, where a, b, c R (a 0). f(x) can be re-written as
Therefore y = f(x) represents a parabola whose axis is parallel to the y-axis, with vertex at
a < 0 and D< 0
f(x) < 0xR.
In this case the parabola remains concave downwards and always below the x-axis.
RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N AND coefficient
A polynomial equation of degree n has n roots (real or imaginary).
If all the coefficients are real then the imaginary roots occur in pairs i.e., number of imaginary roots is always even.
If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.
Polynomial in x of degree n can be factorized into a product of linear/quadratic form.