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Question

The equations x = a open parentheses fraction numerator 1 minus t to the power of 2 end exponent over denominator 1 plus t to the power of 2 end exponent end fraction close parentheses semicolon y equals fraction numerator 2 b t over denominator 1 plus t to the power of 2 end exponent end fraction semicolon t element of R represent

  1. a circle    
  2. an ellipse    
  3. a parabola    
  4. a hyperbola.    

Hint:

Find value of t squared from equation of x and then substitute in equation of y. Simplify  the equation

The correct answer is: an ellipse


     Given :  x space equals a open parentheses fraction numerator 1 minus t squared over denominator 1 plus t squared end fraction close parentheses semicolon y equals fraction numerator 2 b t over denominator 1 plus t squared end fraction semicolon t element of R
    rightwards double arrow x over a space equals space open parentheses fraction numerator 1 minus space t squared over denominator 1 plus space t squared end fraction close parentheses
B y space u sin g space c o m p o n e n d o space minus space d i v i d e n d o space w e space g e t
rightwards double arrow fraction numerator x space plus space a over denominator x space minus a space end fraction space equals space open parentheses fraction numerator left parenthesis 1 minus space t squared right parenthesis space plus space left parenthesis 1 space plus space t squared right parenthesis over denominator left parenthesis 1 minus t squared right parenthesis minus left parenthesis 1 plus space t squared right parenthesis end fraction close parentheses
S i m p l i f y i n g space t h e space a b o v e comma space w e space g e t
rightwards double arrow fraction numerator x space plus space a over denominator x space minus a space end fraction space equals space fraction numerator negative 1 over denominator t squared end fraction
    rightwards double arrow negative t squared space equals space space fraction numerator x space minus space a over denominator x space plus a space end fraction
rightwards double arrow t squared space equals space space fraction numerator a space minus space x over denominator x space plus a space end fraction space minus negative negative negative negative negative space left parenthesis 1 right parenthesis
    Now, consider second equation, 
    rightwards double arrow y space equals fraction numerator 2 b t over denominator 1 space plus space t squared end fraction
S q u a r i n g space b o t h space s i d e s space w e space g e t comma
rightwards double arrow y squared space equals space fraction numerator 4 b squared t squared over denominator left parenthesis 1 space plus space t squared right parenthesis squared end fraction
S u b s t i t u t e space t h e space v a l u e space o f space t squared space i n space t h e space a b o v e space e q u a t i o n comma space w e space g e t
rightwards double arrow y squared space equals space fraction numerator 4 b squared t squared over denominator left parenthesis 1 space plus space t squared right parenthesis squared end fraction space equals space fraction numerator begin display style bevelled fraction numerator 4 b squared over denominator left parenthesis begin display style fraction numerator a minus x over denominator a plus x right parenthesis end fraction end style end fraction end style over denominator left parenthesis 1 space plus space begin display style fraction numerator a space minus x over denominator a space plus x end fraction end style right parenthesis squared end fraction
    O n space f u r t h e r space s i m p l i f y i n g comma space w e space g e t
rightwards double arrow y squared space equals space fraction numerator b squared left parenthesis a minus x right parenthesis left parenthesis a space plus x right parenthesis over denominator a squared end fraction
rightwards double arrow a squared y squared space equals space b squared left parenthesis a minus x right parenthesis left parenthesis a space plus x right parenthesis
rightwards double arrow a squared y squared space equals space b squared left parenthesis a squared minus x squared right parenthesis
rightwards double arrow a squared y squared space equals space b squared a squared minus b squared x squared
rightwards double arrow b squared x squared space plus space a squared y squared space space equals space b squared a squared
D i v i d i n g space b o t h space s i d e s space b y space space b squared a squared comma space w e space g e t
rightwards double arrow x squared over a squared space plus space y squared over b squared space equals space 1 space w h i c h space i s space t h e space g e n e r a l space f o r m space o f space e l l i p s e
    Thus, The equations x = a open parentheses fraction numerator 1 minus t to the power of 2 end exponent over denominator 1 plus t to the power of 2 end exponent end fraction close parentheses semicolon y equals fraction numerator 2 b t over denominator 1 plus t to the power of 2 end exponent end fraction semicolon t element of R represent an ellipse.

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