Equation of Ellipse & Hyperbola: Complex Number

By -
LIKE TO JOIN ON FACEBOOK (click here)

 Equation of Ellipse:

 Ellipse is locus of a point whose sum of distance from two fixed points is constant.
                                                  OR
Ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one focal point to any point of the curve and then back to other focal point has the same length for every point of the curve.

 |z - {z_1}| + |z - {z_2}| = 2a

 |z - z1| + |z - z2| = 2a

 where, 2a > |z1 - z2|

 (if 2a = |z1 - z2| , equation represent a line segment
and if 2a < |z1 - z2|, there is no value of z i.e. sum of distance from two fixed points cannot be lesser than distance between those two fixed points)

 Focii -> z1 and z2

 2a -> Major axis
Ques: |z - 1| + |z + 1| = 4, find the locus of z ?

 Solution: Locus of z is ellipse,

 Focii -> -1 and 1 or (-1,0) and (1,0)

 Major axis, 2a = 4 => a = 2

 therefore, ae = 1,

 => e = 1/2

 since,  b = \sqrt {{a^2}(1 - {e^2})} = \sqrt {4(1 - 1/4)} = \sqrt 3

Since equation of ellipse, \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1

 therefore, \frac{{{x^2}}}{4} + \frac{{{y^2}}}{3} = 1

 or, \frac{{{x^2}}}{4} + \frac{{{y^2}}}{3} = 1
Ques: |z - i| + |z + i| = 3

 Sol: Major axis, 2b = 3

 => b = 3/2

 Focii, i and -i or (0,1) and (0,-1)

 be = 1

 => e = 2/3

 Since, {a^2} = {b^2}(1 - {e^2})

 a = 5/4

 Therefore equation,

 \frac{{4{x^2}}}{5} + \frac{{4{y^2}}}{9} = 1


Equation of Hyperbola:-

 |z - z1| - |z - z2| = 2a

 where, 2a < |z1 - z2|

 Focii, z1 & z2

 LIKE TO JOIN ON FACEBOOK (click here)

Comments

  1. Pranshu8 April 2016 at 23:45

    Thanks a lot man! Was searching for this, definitely helped! :)

    ReplyDelete
  2. gavino13 April 2017 at 05:58

    Thank you.
    And the parabola?

    ReplyDelete
  3. gavino13 April 2017 at 05:58

    Thank you.
    And the parabola?

    ReplyDelete

Post a Comment

Popular posts from this blog

Properties of Modulus & Argument: Complex Number

By -
Image
Properties of Modulus: Modulus of z is the length of vector representing z form origin to the point z. 1.  If z = x + iy then   2.   &  Since,  ( equality follows when y = 0) and   (equality follows when x = 0) IMP 3.  Since, and,  4.  ,  5.  6.  7.   holds when  i.e. both the vectors are in same direction.  holds when   i.e. both the vectors are in opposite direction In general, 8.  IMP 9.  Explanation: Since,  Therefore,  or,                                                  [Since   (? click here ) ] and  10.     ( Property of Parallelogram ) Properties of Argument: Argument is the angle between the vector representing z from the positive direction of x-axis The Principal Argument belongs to (-pi, pi] 1.  ,   2.   , where k = 0, 1 or -1 3.   , where k = 0, 1 or -1 4.   , z cannot be negative Real Number because for negative Real number Arg(z) is equal to pi and so   will become -pi but i
Read more

Fractional Part function

By -
Image
10. Fractional Part f(x) = {x}  Every number 'x' can be written as the sum of its integer and fractional parts. x = [x] + {x} Therefore fractional part, {x} = x - [x] For example, 4.7 = 4 + 0.7 or, 0.7 = 4.7 - 4 i.e. fractional part = Number - Integral Part x = 4.7 , [x] = [4.7] = 4 and {x} = {4.7} = 0.7 Consider, {-4.7} {-4.7} = -4.7 - [-4.7] = -4.7 - (-5)  or, {-4.7} = 0.3 if the number is integer, its fractional part is obviously zero. {6} = 6 - [6] = 6 - 6 = 0 Note- fractional part is always positive and it never becomes 1. in other words, fractional part of a number is the difference between the number and its integral part. Remember,  i.e. fractional part is always less than 1 and greater than or equal to zero. When x will be an integer fractional part will become zero i.e. integer has no fractional part. How to draw the graph, note this, , if  because the fractional part is equal to the number if number is between
Read more

Geometrical Meaning of Argument and Modulus - I: Complex Number

By -
Image
LIKE TO JOIN ON FACEBOOK ( click here ) Students are requested to understand this carefully   1. Arg(z) = pi/3 z = 0 is excluded since z = 0 is the only complex number whose argument is not defined 2. Arg(z) > pi/3 Since principal argument belongs to (-pi, pi],  therefore Argument cannot be greater than pi. therefore Arg(z) > pi/3 =>)  z = 0 is excluded 3. |Arg(z)| < pi/4 => -pi/4 < Arg(z) < pi/4 z = 0 is excluded 4. => Arg(z) = pi since, principal argument belongs to  , therefore Argument cannot be greater than pi. z = 0 is excluded 5.  -pi/3 is included z = 0 is excluded 6. Complex number   , equal only when z is purely real number or '0' IMP 7. Arg(|z|) = 0 Since |z| is a Positive Real number , therefore it lies on right side of x - axis i.e. on positive Real axis and x-axis has '0' argument because it makes 0 radian angle from x-axis For z = 0 , argument does not exist T
Read more