Maths-
General
Easy

Question

The cartesian equation of r equals a s i n space 2 theta is

  1. open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent    
  2. open parentheses x to the power of 2 end exponent minus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent    
  3. open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 2 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent    
  4. open parentheses x to the power of 2 end exponent minus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 2 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent    

Hint:

The term "polar coordinate system" refers to a two-dimensional coordinate system where each point's location on a plane is determined by its distance from a reference point and its angle with respect to a reference direction. Here we have to find the polar equation of r equals a s i n space 2 theta.

The correct answer is: open parentheses x to the power of 2 end exponent plus y to the power of 2 end exponent close parentheses to the power of 3 end exponent equals 4 a to the power of 2 end exponent x to the power of 2 end exponent y to the power of 2 end exponent




    Using the formula, we may generate an endless number of polar coordinates for a single coordinate point.
    (r, +2n) or (-r, +(2n+1)) are possible formulas, where n is an integer.
    If measured in the opposite direction, the value of is positive.
    When calculated counterclockwise, the value of is negative.
    If laid off at the terminal side of, the value of r is positive.
    If r is terminated at the prolongation through the origin from the terminal side of, it has a negative value.
    The equation is given as
    r equals a sin space 2 theta
N o w space w e space k n o w space t h a t colon
sin space 2 theta equals 2 s i n theta cos theta
A p p l y i n g space t h i s comma space w e space g e t colon
r equals 2 a s i n theta cos theta
M u l t i p l y i n g space b y space r squared space o n space b o t h space s i d e s comma space w e space g e t colon
r cubed equals 2 a left parenthesis r s i n theta right parenthesis left parenthesis r cos theta right parenthesis
P u t t i n g space r cos theta equals x space a n d space space r s i n theta equals y comma space w e space g e t colon
r equals square root of x squared plus y squared end root
left parenthesis x squared plus y squared right parenthesis to the power of 3 divided by 2 end exponent equals 2 a y x space
S q u a r i n g space b o t h space s i d e s comma space w e space g e t colon
4 a squared x squared y squared equals left parenthesis x squared plus y squared right parenthesis cubed

    Here we used the concept of the polar coordinate system and also the trigonometric ratios to find the solution. So the equation is 4 a squared x squared y squared equals left parenthesis x squared plus y squared right parenthesis cubed.

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